# 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

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• state variable — būsenos kintamasis statusas T sritis automatika atitikmenys: angl. state variable vok. Zustandsvariable, f rus. параметр состояния, m; переменная состояния, f pranc. variable d état, f …   Automatikos terminų žodynas

• state variable — noun A variable that describes the state of a dynamical system …   Wiktionary

• State variable topology — refers to filters with two pole filter topology. State variable filters are available in lowpass, highpass, bandpass, and notch versions. The notch version is not recommended, because it requires too many amplifiers. It is one of three topologies …   Wikipedia

• 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… …   Wikipedia

• 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. To abstract from the number of inputs, outputs and… …   Wikipedia

• variable d'état — būsenos kintamasis statusas T sritis automatika atitikmenys: angl. state variable vok. Zustandsvariable, f rus. параметр состояния, m; переменная состояния, f pranc. variable d état, f …   Automatikos terminų žodynas

• Variable universal life insurance — (often shortened to VUL) is a type of life insurance that builds a cash value. In a VUL, the cash value can be invested in a wide variety of separate accounts, similar to mutual funds, and the choice of which of the available separate accounts to …   Wikipedia

• Variable-order Markov model — Variable order Markov (VOM) models are an important class of models that extend the well known Markov chain models. In contrast to the Markov chain models, where each random variable in a sequence with a Markov property depends on a fixed number… …   Wikipedia

• Variable retention — is a relatively new silvicultural system that follows nature s model by always retaining a significant part of the forest after harvesting. Variable retention harvests serve to promote desired natural regeneration while preserving unique features …   Wikipedia

• Variable displacement — is an automobile engine technology that allows the engine displacement to change, usually by deactivating cylinders, for improved fuel economy. The technology is primarily used in large, multi cylinder engines. Many automobile manufacturers have… …   Wikipedia