A state variable is one of the set of variables that describe the "state" of a dynamical system

Dynamical system

A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

. 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

Mechanics

Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....

, 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

Thermodynamics

Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

, state function

State function

In thermodynamics, a state function, function of state, state quantity, or state variable is a property of a system that depends only on the current state of the system, not on the way in which the system acquired that state . A state function describes the equilibrium state of a system...

s are properties such as temperature

Thermodynamic temperature

Thermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. Thermodynamic temperature is an "absolute" scale because it is the measure of the fundamental property underlying temperature: its null or zero point, absolute zero, is the...

, pressure

Pressure

Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

, volume

Volume

Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

, internal energy

Internal energy

In thermodynamics, the internal energy is the total energy contained by a thermodynamic system. It is the energy needed to create the system, but excludes the energy to displace the system's surroundings, any energy associated with a move as a whole, or due to external force fields. Internal...

, enthalpy

Enthalpy

Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...

, and entropy

Entropy

Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

.

In ecosystem model

Ecosystem model

An ecosystem model is an abstract, usually mathematical, representation of an ecological system , which is studied to gain a deeper understanding of the real system.Ecosystem models are formed by combining known ecological relations An ecosystem model is an abstract, usually mathematical,...

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

Control Systems Engineering

In Control Engineering

Control engineering

Control engineering or Control systems engineering is the engineering discipline that applies control theory to design systems with predictable behaviors...

 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

State space (controls)

In control engineering, a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations...

 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

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

:

existing in dimension R^N*N

existing in dimension R^N*L

existing in dimension R^M*N

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 , where n is the discrete

Discrete time

Discrete time is the discontinuity of a function's time domain that results from sampling a variable at a finite interval. For example, consider a newspaper that reports the price of crude oil once every day at 6:00AM. The newspaper is described as sampling the cost at a frequency of once per 24...

 point at which the system is being evaluated. The discrete-time state equations are

, which describes the next state of the system (x(n+1)) with respect to current state and inputs u(n) of the system.

, 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 , and the continuous time state equations are

, which describes the next state of the system with respect to current state x(t) and inputs u(t) of the system.

, 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)
  State space (controls)
  In control engineering, a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations...
  
  
• Control theory
  Control theory
  Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
  
  
• Equation of state
  Equation of state
  In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...
  
  
• State (computer science)
  State (computer science)
  In computer science and automata theory, a state is a unique configuration of information in a program or machine. It is a concept that occasionally extends into some forms of systems programming such as lexers and parsers....
  
  
• State (physics)
• State (functional analysis)
  State (functional analysis)
  In functional analysis, a state on a C*-algebra is a positive linear functional of norm 1. The set of states of a C*-algebra A, sometimes denoted by S, is always a convex set. The extremal points of S are called pure states...
  
  
• State diagram
  State diagram
  A state diagram is a type of diagram used in computer science and related fields to describe the behavior of systems. State diagrams require that the system described is composed of a finite number of states; sometimes, this is indeed the case, while at other times this is a reasonable abstraction...
  
  
• State variable filter
  State variable filter
  A state variable filter is a type of active filter. It consists of one or more integrators, connected in some feedback configuration. Any LTI system can be described as a state-space model, with n state variables for an nth-order system. A state variable filter realizes the state-space model directly...