Feedback Control & Dynamical Systems

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Abstractions about modeling and steering dynamical systems through state, feedback, transfer functions, stability, and optimization. They include PID and sliding-mode control, linearization, observability tests, Lyapunov methods, trajectory design, and state-space representation.

29 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Backstepping — A recursive nonlinear-control design method for strict-feedback systems that constructs a stabilizing controller and Lyapunov function stage by stage.
• Capacitor-spring analogy — Map an ideal electrical capacitor to a mechanical spring by preserving constitutive integral form, with voltage corresponding to force, current to velocity, charge to displacement, and capacitance to compliance.
• Causal loop diagram — Map hypothesized causal influence among changing variables with signed arrows and closed reinforcing or balancing loops, providing a qualitative feedback model whose links require narrative, boundary, and evidence.
• Closed-loop transfer function — Collapse a linear feedback interconnection into the net input-to-output map—typically G(s)/(1+G(s)H(s)) for negative feedback—whose denominator exposes stability, sensitivity, and loop-gain effects.
• Correlation integral — The probability, estimated from state pairs, that two independently sampled points on a trajectory or invariant measure lie within a specified distance.
• Covector mapping principle — A compatibility principle giving conditions under which discretizing an optimal-control problem and then dualizing yields covectors corresponding to a discretization of the continuous Pontryagin adjoint system.
• Dead-beat control — A discrete-time control design that drives a controllable system's state or output exactly to its target in the minimum finite number of sampling steps.
• Feedback linearization — A nonlinear-control technique that uses state or output transformations and a compensating input law to cancel modeled nonlinearities and expose linear closed-loop dynamics.
• Flatness (systems theory) — A nonlinear-system property in which all states and inputs can be parameterized by a flat output and finitely many of its derivatives, without integrating differential equations.
• Full state feedback — A control design that feeds a measured or estimated state vector through a gain matrix to place a controllable linear system’s closed-loop poles.
• H-infinity loop-shaping — A robust-control design that first frequency-shapes a plant and then optimizes a stabilizing controller against normalized coprime-factor uncertainty.
• Hamiltonian (control theory) — The function combining instantaneous objective and state dynamics through costate variables in optimal control, whose pointwise optimization is required by Pontryagin's maximum principle.
• Hautus lemma — A rank-test lemma characterizing controllability, observability, stabilizability, and detectability of linear time-invariant state-space systems at eigenvalues.
• Impulse vector — A phasor-like graphical representation of an input-shaper impulse's amplitude, phase and residual-vibration contribution, used to construct sequences whose vectors cancel.
• Kalman–Yakubovich–Popov lemma — A theorem equating a frequency-domain positivity condition for a linear system with existence of a state-space quadratic certificate.
• Linear dynamical system — A dynamical system whose state evolution and output laws are linear, enabling superposition and analysis through matrices, spectra, and modes.
• Lyapunov redesign — A nonlinear-control method that augments a nominal stabilizing feedback law using a known Lyapunov function to preserve stability under matched uncertainty.
• Optimal control — The selection of a time-dependent control policy for a dynamical system that minimizes or maximizes an objective while satisfying dynamics and constraints.
• PID controller — A feedback controller that combines proportional response to current error, integral response to accumulated error and derivative response to error trend.
• Proper transfer function — A rational transfer function whose numerator degree does not exceed its denominator degree, so high-frequency gain remains finite.
• Rosenbrock system matrix — A polynomial block matrix combining state-space dynamics and input-output equations of a linear system.
• Separation principle — A control-theory result allowing state estimation and feedback control to be designed independently while preserving stability or optimality under stated linear-system assumptions.
• Servo (radio control) — A compact closed-loop actuator module that converts a command pulse or digital signal into controlled shaft or linkage position.
• Sliding mode control — A nonlinear variable-structure control method using discontinuous feedback to drive trajectories onto a designed switching manifold and maintain reduced-order motion along it.
• State variable — One coordinate in a minimal sufficient state description whose current values, together with inputs and a model, determine the system's admissible future evolution and observable outputs.
• State-transition matrix — A matrix function Φ(t,t₀) mapping the state of a linear dynamical system at time t₀ to its homogeneous state at time t.
• Strange nonchaotic attractor — An invariant attracting set with geometrically nonsmooth or fractal structure but no positive maximal Lyapunov exponent.
• Trajectory optimization — The computation of a state-and-control path that extremizes a performance objective while satisfying dynamics, boundary conditions and path constraints, usually as an open-loop optimal-control solution.
• Weighting pattern — Represent a linear time-varying system's zero-state input–output action by a two-time kernel formed from output, state-transition, and input maps, reducing to an impulse-response convolution kernel in the time-invariant case.