State variable

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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 In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
. 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 equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
s are said to be in state-variable form.

Examples

*In
mechanical systems A machine is a physical system using power to apply forces and control movement to perform an action. The term is commonly applied to artificial devices, such as those employing engines or motors, but also to natural biological macromolecul ...
, the position coordinates and
velocities Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is ...
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 an independent variable of a
state function In the thermodynamics of equilibrium, a state function, function of state, or point function for a thermodynamic system is a mathematical function relating several state variables or state quantities (that describe equilibrium states of a system ...
. Examples include
internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kine ...
,
enthalpy Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant p ...
, temperature, pressure, volume and entropy. Heat and
work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking ** Working animal, an animal ...
are not state functions, but
process function In thermodynamics, a quantity that is well defined so as to describe the path of a process through the equilibrium state space of a thermodynamic system is termed a process function, or, alternatively, a process quantity, or a path function. As ...
s. *In
electronic Electronic may refer to: * Electronics, the science of how to control electric energy in semiconductor * ''Electronics'' (magazine), a defunct American trade journal *Electronic storage, the storage of data using an electronic device *Electronic c ...
/
electrical circuit An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sources, ...
s, the voltages of the nodes and the currents through components in the circuit are usually the state variables. In any electrical circuit, the number of state variables are equal to the number of (independent) storage elements, which are inductors and capacitors. The state variable for an inductor is the current through the inductor, while that for a capacitor is the voltage across the capacitor. *In
ecosystem model An ecosystem model is an abstract, usually mathematical, representation of an ecological system (ranging in scale from an individual population, to an ecological community, or even an entire biome), which is studied to better understand the r ...
s, population sizes (or concentrations) of plants, animals and resources (nutrients, organic material) are typical state variables.

Control systems engineering

In
control engineering Control engineering or control systems engineering is an engineering discipline that deals with control systems, applying control theory to design equipment and systems with desired behaviors in control environments. The discipline of controls ...
and other areas of science and engineering, state variables are used to represent the states of a general system. The set of possible combinations of state variable values is called the
state space A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory. For instance, the to ...
of the system. The equations relating the current state of a system to its most recent input and past states are called the state equations, and the equations expressing the values of the output variables in terms of the state variables and inputs are called the output equations. As shown below, the state equations and output equations for a linear time invariant system can be expressed using coefficient
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
: ''A'', ''B'', ''C'', and ''D'' :$A \in \R^, \quad B \in \R^, \quad C \in \R^, \quad D \in \R^ ,$ where ''N'', ''L'' and ''M'' are the dimensions of the vectors describing the state, input and output, respectively.

Discrete-time systems

The state vector (vector of state variables) representing the current state of a
discrete-time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "po ...
system (i.e. digital system) is

Continuous time systems

The state vector representing the current state of a continuous-time system (i.e. analog system) is $x\left(t\right)$, and the continuous-time state equations giving the evolution of the state vector are : $\frac = Ax\left(t\right) + Bu\left(t\right),$ which describes the continuous rate of change $\frac$ of the state of the system with respect to current state ''x''(''t'') and inputs ''u''(''t'') of the system. The output equations are : $y\left(t\right) = Cx\left(t\right) + Du\left(t\right),$ which describes the output ''y''(''t'') with respect to current states ''x''(''t'') and inputs ''u''(''t'') to the system.

* State space (controls) * Control (optimal control theory) *
Control theory Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
* Equation of state *
State (computer science) In information technology and computer science, a system is described as stateful if it is designed to remember preceding events or user interactions; the remembered information is called the state of the system. The set of states a system can ...
*
Dynamical systems In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
*
State (functional analysis) In functional analysis, a state of an operator system is a positive linear functional of norm 1. States in functional analysis generalize the notion of density matrices in quantum mechanics, which represent quantum states, both . Density matric ...
*
State diagram A state diagram is a type of diagram used in computer science and related fields to describe the behavior of systems. State diagrams require that the system described is composed of a finite number of states; sometimes, this is indeed the case ...
*
State variable filter A state variable filter is a type of active filter in electronic circuits. It consists of one or more integrators, connected in some feedback configuration. It is essentially used when precise Q factor is required, as other multi-order filters are ...

References

{{DEFAULTSORT:State Variable Classical control theory