A state-space model is a mathematical model of a controlled system as a set of input, output and state variables related by first-order differential equations.  To abstract from the number of inputs, outputs and states, the variables are expressed in vectorized form and the algebraic and differential equations are written in matrix form. The state-space model provides a simple and compact way to model and examine Multi-Input Multi-Output systems.  We would otherwise have to write down Laplace transforms to encode all the states of a system.  Unlike the frequency domain analysis methods, the use of the state-space representation is not limited to systems with only linear components and zero initial conditions. "State-space" refers to the space whose axes are the state variables. The states of the mechanism can be modeled as in vectorized form within that space. 

State Variables

The internal state variables are the minimum possible subset of system variables that can represent the entire state of the system at any given time.  State equations must be linearly independent.  The smallest number of state equations required to represent a given system, n, is usually equal to the size of the model's defining differential equations.  If the system is represented in Laplace (s-domain) form, the least number of state variables is equal to the size of the transfer function's denominator after it has been reduced to a proper fraction.  It is important to appreciate that changing a state-space realization to a transfer function form will sometimes lose internal information about the system, and can provide a description of a system which is stable, when the state-space realization is unstable at times. 

State space Compensator Analysis

In many Single-Input Single Output systems a compensator can be designed through a method named ''Loop Shaping'' using Frequency Domain methodologies; the Bode plot is the primary tool used for ''Loop Shaping''.  For Multi-Input Multi-Output mechanisms the singular value decomposition (SVD) is a tool similar to the Bode plot.  ''Loop Shaping'' is more difficult for Multi-Input Multi-Output mechanisms than SISO. 

Since ''Loop Shaping'' is diffcult for Multi-Input Multi-Output systems other techniques were developed for creating state-space controllers.  The two most popular state space controllers are the Linear Quadratic Regulator (LQR controller) and the Linear Quadratic Gaussian (LQG).