Vector-Matrix Algebra and the State-Space Representation of Dynamic Systems

Abstract

An alternative to deriving a high-order differential system equation is to represent the system as a set of simultaneous first-order differential equations of the system’s state variables, which are its energy storage variables. This is known as state-space. State equations are solved numerically, using finite difference approximations. They can also be solved using Laplace transformations to yield transfer functions. State-space has three advantages. It is much easier to derive a set of state equations than the corresponding high-order differential equation. It is also easy to include non-linear properties, since the solution uses numerical methods. Finally, state-space is the basis of Modern Control theory, which enables the control designer to arbitrarily tailor the response of a system, within the physical limitations of the hardware.