Fundamentals of Dynamical Systems

For the purposes of control system design, analysis, test, and repair, the most important part of the very broad subject known as system theory is the theory of dynamical systems. It is difficult to give a precise and sufficiently general definition of a dynamical system for reasons that will become evident from the detailed discussion to follow. All systems that can be described by ordinary differential or difference equations with real coefficients (ODEs) are indubitably dynamical systems. A very important example of a dynamical system that cannot be described by a continuous-time ODE is a pure delay. Most of this chapter will deal with different ways to describe and analyze dynamical systems. We will precisely specify the subclass of such systems for which each description is valid.

The idea of a system involves an approximation to reality. Specifically, a system is a device that accepts an input signal and produces an output signal. It is assumed to do this regardless of the energy or power in the input signal and independent of any other system connected to it. Physical devices do not normally behave this way. The response of a real system, as opposed to that of its mathematical approximation, depends on both the input power and whatever load the output is expected to drive.