# State variable

A state variable is one of the set of variables that are used to describe the mathematical "state" of a dynamical system. Intuitively, the state of a system describes enough about the system to determine its future behaviour in the absence of any external forces affecting the system. Models that consist of coupled first-order differential equations are said to be in state-variable form.[1]

## Control systems engineering

In control engineering and other areas of science and engineering, state variables are used to represent the states of a general system. The state variables can be used to describe the state space of the system. The equations relating the current state and output of a system to its current input and past states are called the state equations. The state equations for a linear time invariant system can be expressed using coefficient matrices:

$A\in$ RN*N, $\quad B\in$ RN*L, $\quad C\in$ RM*N, $\quad D\in$ RM*L,

where N, L and M are the dimensions of the vectors describing the state, input and output, respectively.

### Discrete-time systems

The state variable representing the current state of a discrete-time system (i.e. digital system) is $x[n]\,$, where n is the discrete point at which the system is being evaluated. The discrete-time state equations are

$x[n+1] = Ax[n] + Bu[n]\,\!$ , which describes the next state of the system (x[n+1]) with respect to current state and inputs u[n] of the system.
$y[n] = Cx[n] + Du[n]\,\!$ , which describes the output y[n] with respect to current states and inputs u[n] to the system.

### Continuous time systems

The state variable representing the current state of a continuous-time system (i.e. analog system) is $x(t)\,$, and the continuous time state equations are

$\frac{dx(t)}{dt} \ = Ax(t) + Bu(t)\,\!$ , which describes the next state of the system $\frac{dx(t)}{dt} \,\!$ with respect to current state x(t) and inputs u(t) of the system.
$y(t) = Cx(t) + Du(t)\,\!$ , which describes the output y(t) with respect to current states x(t) and inputs u(t) to the system.