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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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155 news items


Thu, 03 Sep 2015 02:00:00 -0700

In those cases, simply changes in a state variable such as leg compression or perturbations in CoM height may induce grounded running to change to mixed gaits or vice versa. One example is presented in Fig. 4. There, both solutions exhibit similar ...


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The National Law Review

The National Law Review
Mon, 24 Aug 2015 16:02:02 -0700

Turning to the asserted grounds of unpatentability, the Board then considered the proper scope of the claim terms “meta-rights,” “state variable,” and “repository.” With respect to “meta-rights” and “state variable,” the Board concluded that the terms ...
Thu, 13 Aug 2015 10:01:36 -0700

Imagine a robot with an all-around bump sensor. The response to the bump sensor activating depends on the previous state of the robot. If it had been going forward, a bump will send it backwards and vice versa. This robot exhibits behavior that is easy ...
EDN.com (blog)
Thu, 20 Aug 2015 16:23:19 -0700

For example, a state variable could be implemented as part of the expression passed to ASSERT. But if the expression passed to ASSERT has side effects, that is, it changes the state of the embedded system, then disabling assertions will change the ...
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Both the IDT and the PaG&E settlements would impose a more than one-year moratorium on selling variable rates in the state. Variable rates allow consumer bills to change month to month based on fluctuations in the market. The companies would be ...
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With this vote, the state's variable toll lane project got its first bump up the chain, but there are many votes yet to take place before one spoonful of dirt is turned. Funding from the Legislature to the tune of $3.4 billion must be allocated, and I ...
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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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