# State variable

State variable

A state variable is an element of the set of variables that describe the state of a dynamical system.

In case of simple mechanical systems, position coordinates and their derivates are typical state variables. Temperature, pressure, internal energy, enthalpy, entropy are examples of state variables in a thermodynamics system.

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 (controls) 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 are expressed with Coefficient matrices:

$A,!$ existing in dimension R"N*N"

$B,!$ existing in dimension R"N*L"

$C,!$ existing in dimension R"M*N"

$D,!$ existing in dimension R"M*L"

Discrete-Time Systems

The state variable representing the current state of a discrete-time system (i.e. digital systems) is $x\left(n\right),$, where n is the discrete point at which the system is being evaluated. The discrete-time state equations are : $x\left(n+1\right) = Ax\left(n\right) + Bu\left(n\right),!$ , which describes the next state of the system (x(n+1)) with respect to current state and inputs u(n) of the system.

: $Y\left(n\right) = Cx\left(n\right) + Du\left(n\right),!$ , which describes the output Y(n) with respect to current states and inputs u(n) to the system.

Continuous Time Systems (Analog)

The state variable representing the current state of a continuous-time system (i.e. analog systems) is $x\left(t\right),$, and the continuous time state equations are : $frac\left\{dx\left(t\right)\right\}\left\{dt\right\} = Ax\left(t\right) + Bu\left(t\right),!$ , which describes the next state of the system $frac\left\{dx\left(t\right)\right\}\left\{dt\right\} ,!$ with respect to current state x(t) and inputs u(t) of the system.

: $Y\left(t\right) = Cx\left(t\right) + Du\left(t\right),!$ , which describes the output Y(t) with respect to current states x(t) and inputs u(t) to the system.

ee also

*State space (controls)
*Control theory
*Equation of state
*State (computer science)
*State (physics)
*State (functional analysis)
*State diagram
*State variable filter

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