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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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214 news items

Sun, 17 Apr 2016 08:17:42 -0700

using a flag or state variable to flee at 50 and stop fleeing at 75 (or whatever) results in oscillations back and forth at ranges of 50 and 75 feet. is one check per possible move the best approach, or should i go to a lower level and prevent ...
Mon, 15 Dec 2014 00:42:27 -0800

The usual schematic of a state-variable filter with two inverting integrators is well known. Curiously, the input signal is almost always connected to the minus input of U1. Figure 1 is an example with ω0 = 1kHz and Q = 5. Figure 1 Typical state ...


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The KOMA Elektronik SVF-201 is a state variable filter for the Eurorack modular synthesizer system. Its filter circuit is built up with optical parts (Vactrols) therefore delivering excellent studio audio quality and an unique warm analogue sound. The ...


Wed, 03 Oct 2012 04:26:15 -0700

In introducing their new Q4, Radial has debuted what they say “could very well be the world's first” 100% discrete state-variable class-A parametric equalizer. This 500-series module appears to be an enticing mix of classic engineering and new design.

Columbus Dispatch

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Tickets for the 2016 home game against Michigan are expected to cost a record $195 under Ohio State's variable-pricing model, according to Columbus Business First. The Ohio State Board of Trustees will consider the athletic council's recommendation for ...

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At least Rogers, as a native son with Cherokee ancestors, knew firsthand about the state's variable weather. As 2015 draws to a close, we realize the weather this year has been changeable, to say the least. Good examples are last May, which ...

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A recurrent spiking neural network is proposed that implements planning as probabilistic inference for finite and infinite horizon tasks. The architecture splits this problem into two parts: The stochastic transient firing of the network embodies the ...

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