Linear dynamical system

A system whose state evolves according to linear equations, making its response to combined inputs the corresponding combination of individual responses.

State-space representation: A description of a dynamical system using state variables, inputs, outputs, and equations connecting them. It expresses a linear dynamical system directly through its state and input equations.
Linear map: A function between vector spaces that preserves addition and scalar multiplication. The system’s equations use linear maps to relate states, inputs, and outputs.
Control theory: The study of how to influence dynamical systems to achieve desired behavior. Linear system models support analysis and design of controllers.
Nonlinear system: A system whose governing equations do not satisfy linearity. Nonlinear systems generally do not obey superposition.
Lyapunov stability: A stability property in which sufficiently small initial perturbations remain small over time. It distinguishes bounded state evolution from unstable growth.
State-transition matrix: A matrix that maps a linear system’s initial state to its later state in the absence of inputs. It captures how the system’s internal state changes over time.
Ordinary differential equation: An equation relating an unknown function of one variable to its derivatives. Continuous-time linear systems are often specified by differential equations.
Kalman filter: An algorithm that estimates a changing state from noisy measurements using a linear model and probabilistic assumptions. Its standard form estimates states in linear dynamical systems with noise.
Time-varying system: A system whose governing rules explicitly change with time. A linear system can still be time-varying; fixed coefficients are an additional assumption.
Controllability: The property that inputs can drive a system from one state to another within a specified time. A linear model permits algebraic tests of which states inputs can reach.

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Characteristic polynomialRelated: Characteristic roots reveal growth, decay, and oscillation modes.