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Introduction
Control theory is a branch of engineering that deals with the control of en-
gineering systems. The engineering systems could be from diverse domains,
and the applications include controlling the power output of an automotive
engine, stabilizing the rate of rotation of an electric motor, and controlling
the “rate of reaction” (or the speed of a chemical process). The control of
these systems is exercised by the manipulation of so-called control variables.
For example, in the case of controlling the power output of an automotive
engine, the control variable could be the amount of fuel injected into the en-
gine. This would control the thrust of the piston and therefore the power
output from the engine. Similarly, in the other two examples, the control
variables could be the amount of current flowing through the motor coils or
the ambient temperature of a chemical process. Thus, the control variables
provide a harness that helps us to control the system effectively.
Prior to the proposal of Kalman filtering, the typical approach for sys-
tem control involved the specification of a fully comprehensive mathemat-
ical model describing the system dynamics. The model is usually formulated
in the form of a differential equation. This helps to determine in a quanti-
tative manner the effects of the control variables on system dynamics. Con-
trol is then effected by manipulating the variables as prescribed by the
model.
Along with the preceding approach also came a painful realization of its
limitations. The mathematical models may not be 100-percent accurate, as
there may be some approximations used in the modeling process. Addition-
ally, the instruments used to measure the system parameters may have some
built-in inaccuracies that could result in measurement error. To compound
things even further, there may also be some extraneous disturbances to the
system that cannot be anticipated and modeled in a deterministic fashion.
Hence, the aforementioned methodology becomes increasingly harder to
implement as the systems grow in complexity.
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