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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]}
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Examples[edit]
- 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[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:
- R^{N*N}, R^{N*L}, R^{M*N}, R^{M*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 , where n is the discrete point at which the system is being evaluated. The discrete-time state equations are
- , which describes the next state of the system (x[n+1]) with respect to current state and inputs u[n] of the system.
- , 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 , and the continuous time state equations are
- , which describes the next state of the system with respect to current state x(t) and inputs u(t) of the system.
- , which describes the output y(t) with respect to current states x(t) and inputs u(t) to the system.
See also[edit]
- State space (controls)
- Control theory
- Equation of state
- State (computer science)
- Dynamical systems
- State (functional analysis)
- State diagram
- State variable filter
References[edit]
- ^ William J. Palm III (2010). System Dynamics (2nd ed.). p. 225.
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