State variable

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]

Examples

• In mechanical systems, the position coordinates and velocities of mechanical parts are typical state variables; knowing these, it is possible to determine the future state of the objects in the system.
• In thermodynamics, a state variable is also called a state function. Examples include temperature, pressure, volume, internal energy, enthalpy, and entropy. In contrast heat and work are not state functions, but process functions.
• In electronic circuits, the voltages of the nodes and the currents through components in the circuit are usually the state variables.
• In ecosystem models, population sizes (or concentrations) of plants, animals and resources (nutrients, organic material) are typical state variables.

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 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:

$A\in$ R^N*N, $\quad B\in$ R^N*L, $\quad C\in$ R^M*N, $\quad D\in$ R^M*L,

where N, L and M are the dimensions of the vectors describing the state, input and output, respectively.

Discrete-time systems

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

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

• State space (controls)
• Control theory
• Equation of state
• State (computer science)
• Dynamical systems
• State (functional analysis)
• State diagram
• State variable filter

References

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