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.

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 set of possible combinations of state variable values is called the state space of the 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:

$A\in$  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 vector (vector of state variables) representing the current state of a discrete-time system (i.e. digital system) is $x[n]\,$ , where n is the discrete point in time 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. The output equations are

$y[n]=Cx[n]+Du[n],\,\!$

which describes the output y[n] with respect to current states and inputs u[n] to the system.

Continuous time systems

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

${\frac {dx(t)}{dt}}\ =Ax(t)+Bu(t),\,\!$

which describes the continuous rate of change ${\frac {dx(t)}{dt}}\,\!$  of the state of the system with respect to current state x(t) and inputs u(t) of the system. The output equations are

$y(t)=Cx(t)+Du(t),\,\!$

which describes the output y(t) with respect to current states x(t) and inputs u(t) to the system.