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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:

${\displaystyle A\in }$ RN*N, ${\displaystyle \quad B\in }$ RN*L, ${\displaystyle \quad C\in }$ RM*N, ${\displaystyle \quad D\in }$ 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 ${\displaystyle x[n]\,}$, where n is the discrete point at which the system is being evaluated. The discrete-time state equations are

${\displaystyle x[n+1]=Ax[n]+Bu[n]\,\!}$ , which describes the next state of the system (x[n+1]) with respect to current state and inputs u[n] of the system.
${\displaystyle y[n]=Cx[n]+Du[n]\,\!}$ , 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 ${\displaystyle x(t)\,}$, and the continuous time state equations are

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

## References

1. ^ William J. Palm III (2010). System Dynamics (2nd ed.). p. 225.

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