State space

A state space is the set of all possible states of a system, with each state represented as a point. Its structure captures which configurations the system can occupy.

State (dynamical systems): A complete description of a system’s configuration at a particular time. Each point in a state space encodes one possible system state.
State-space representation: A mathematical form that describes a dynamic system using state variables and their evolution equations. It makes state-space points and trajectories explicit in system models.
Control theory: The study of how inputs can guide the behavior of dynamic systems. State-space models let controllers account for coupled variables and changing system states.
Dynamical system: A system whose state changes over time according to a rule or evolution law. Its possible states and evolution rules together define the setting for state-space analysis.
Phase space: A state space whose coordinates commonly include positions and momenta or velocities. It is the standard state-space representation for mechanical systems.
State transition: A change from one state of a system to another under its dynamics or an input. Transitions trace how a system moves between points in its state space.
Linear state-space model: A dynamic system model in which state evolution and outputs are linear functions of states and inputs. It is a widely used state-space form for engineering analysis and design.
State space (probability): The set of possible outcomes or configurations represented in a probabilistic model. Probability uses state spaces to define events and distributions, not necessarily physical trajectories.
Configuration space: The space of a system’s possible positions or configurations, without necessarily including velocities. It tracks configurations alone, while a full state space may also encode motion.
Trajectory (dynamical systems): The curve or sequence of states that a system visits as it evolves over time. A trajectory is the path drawn through state space by one system history.

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