State variable

A state variable is one of the set of variables that describe the "state" of a dynamical system. Intuitively, the state of a system describes enough about the system to determine its future behaviour. Models that consist of coupled first-order differential equations are said to be in state-variable form

Examples

In simple mechanical systems, position coordinates and their derivatives are typical state variables; knowing these, it is possible to determine the future state of the objects in the system.

In a thermodynamic system, state functions are properties such as temperature, pressure, volume, internal energy, enthalpy, and entropy.

In ecosystem models, population sizes (or concentrations) of plants, animals and resources (nutrients, organic material) are typical state variables.

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 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 $x(n)\,$, 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 (Analog)

The state variable representing the current state of a continuous-time system (i.e. analog systems) 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.

See also

• State space (controls)
• Control theory
• Equation of state
• State (computer science)
• Dynamical systems
• State (functional analysis)
• State diagram
• State variable filter