Models that consist of coupled first-order differential equations are said to be in state-variable form.
[1] In control engineering and other areas of science and engineering, state variables are used to represent the states of a general system.
The equations relating the current state of a system to its most recent input and past states are called the state equations, and the equations expressing the values of the output variables in terms of the state variables and inputs are called the output equations.
As shown below, the state equations and output equations for a linear time invariant system can be expressed using coefficient matrices: A, B, C, and D where N, L and M are the dimensions of the vectors describing the state, input and output, respectively.
, where n is the discrete point in time at which the system is being evaluated.