STATE-DETERMINED SYSTEMS 9
1.3 STATE-DETERMINED SYSTEMS
The goal of this book is to describe means for setting up mathematical mod-
els for systems. The type of model that will be found is often described as a
state-determined system. In mathematical notation, such a system model is often
described by a set of ordinary differential equations in terms of so-called state
variables and a set of algebraic equations that relate other system variables
of interest to the state variables. In succeeding chapters an orderly procedure,
beginning with physical effects to be modeled and ending with state differential
equations, will be demonstrated. Even though some techniques of analysis and
computer simulation do not require that the state equations be written explictly,
from a mathematical point of view all the system models that will be discussed
are state-determined systems.
The future of all the variables associated with a state-determined system can
be predicted if (1) the state variables are known at some initial time and (2) the
future time history of the input quantities from the external environment is known.
Such models, which are virtually the only types used in engineering, have
some built-in philosophical implications. For example, events in the future do
not affect the present state of the system. This implication is correlated with
the assumption that time runs only in one direction—from past to future. That
models should have these properties probably seems plausible, if not obvious,
yet it is remarkably difficult to conceive of a demonstration that real systems
always have these properties.
Clearly, past history can have an effect on a system; yet the influence of
the past is exhibited in a special way in state-determined systems. All the past
history of a state-determined system is summed up in the present values of its
state variables. This means that many past histories could have resulted in the
same present value of state variables and hence the same future behavior of
the system. It also means that if one can condition the system to bring the state
variables to some particular values, then the future system response is determined
by the future inputs and nothing is important about the past except that the state
variables were brought to those values.
Scientific experiments are run as if the systems to be studied were state deter-
mined. The system is always started from controlled conditions that are expressed
in terms of carefully monitored variables. If the experiment is repeatable, then
the assumption is that the state variables are properly initialized by the opera-
tions used to set up the experiment. If the experiment is not repeatable, then the
assumption is that some important influence has not been controlled. This influ-
ence can be either a state variable that was not monitored and initialized properly
or an unrecognized input quantity through which the environment influences the
system.
State-determined system models have proved useful over centuries of sci-
entific and technical work. For the usual macroscopic systems encountered in
engineering, state-determined system models are nearly universal, and there is
continuing interest in developing such models for social and economic systems.
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