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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[edit]

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:

A\in RN*N, \quad B\in RN*L, \quad C\in RM*N, \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[edit]

The state variable representing the current state of a discrete-time system (i.e. digital system) 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[edit]

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


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

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Mon, 27 Jul 2015 08:56:15 -0700

where γ is the state variable, σ is the effective normal stress, A is a fault constitutive parameter and S is the coulomb stress function. The starting state variable is , where is the background stressing rate. The earthquake rate R at the midpoint of ...

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Thu, 23 Jul 2015 03:41:15 -0700

Soil moisture is a primary state variable of hydrology and the water cycle on land, and is an initial or boundary condition of Earth science models. An assessment of surface soil moisture is necessary for applications such as weather forecasting ...
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Sun, 05 Jul 2015 09:30:00 -0700

The Board also noted that Gruse fails to disclose at least one state variable that identifies a location where a state of rights is tracked and associating the at least one state variable with the at least one right in the first location. The Petition ...
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Fri, 17 Jul 2015 04:50:56 -0700

Including electron spin as an additional state variable offers new prospects for information processing, enabling future non-volatile, reprogrammable devices beyond the current semiconductor technology roadmap. Silicon is an ideal host for such a spin ...


Tue, 14 Jan 2014 12:08:49 -0800

UNIVERSITY PARK, Pa. - Penn State Athletics today announced variable pricing structure for its single game public football tickets during the 2014 season. Variable pricing determines individual game prices from many factors including expected demand ...


Wed, 25 Feb 2015 11:16:27 -0800

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