Feedback Control Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Feedback Control Ontology within the Kireth Research Semantic Library. Feedback control studies closed-loop systems in which measured or estimated output or state is compared with a reference and the resulting error or state information is used to modify future control action. This ontology supports negative and positive feedback, reference tracking, regulation, disturbance rejection, sensitivity reduction, closed-loop transfer functions, loop gain, stability margins, PID control, state feedback, observer-based feedback, frequency response, robustness, actuator limits, noise sensitivity and relationships with control theory, dynamical systems, signal processing, estimation and optimisation.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Feedback Control Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Control Theory / Feedback Control"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/control-theory/feedback-control-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = FeedbackControl
 SYMBOLIC_FORMS = [
  "e(t)=r(t)-y(t)",
  "u(t)=C[e(t)]",
  "T(s)=G(s)C(s)/(1+G(s)C(s))"
 ]
 ENTITY_CLASS = ClosedLoopControlFramework
 PRIMARY_CONTEXT = ControlTheory
 DEFINITION = "Feedback control is the use of measured or estimated system behaviour to modify control action so that closed-loop performance approaches a desired objective."
 SEMANTIC_ROLE = "Provides the machine-readable framework for closed-loop error correction, reference tracking, regulation, disturbance rejection, sensitivity shaping, stability and robust feedback design."
}

SCOPE {
 INCLUDE = [
  FeedbackControl,
  FeedbackLoop,
  NegativeFeedback,
  PositiveFeedback,
  ReferenceSignal,
  MeasuredOutput,
  ControlledOutput,
  ErrorSignal,
  Controller,
  Plant,
  Sensor,
  Actuator,
  ControlInput,
  LoopGain,
  OpenLoopTransferFunction,
  ClosedLoopTransferFunction,
  SensitivityFunction,
  ComplementarySensitivityFunction,
  DisturbanceRejection,
  NoiseRejection,
  ReferenceTracking,
  Regulation,
  Stability,
  ClosedLoopStability,
  GainMargin,
  PhaseMargin,
  BodePlot,
  NyquistCriterion,
  RootLocus,
  ProportionalControl,
  IntegralControl,
  DerivativeControl,
  PIDControl,
  StateFeedback,
  OutputFeedback,
  ObserverBasedFeedback,
  FeedforwardCompensation,
  RobustFeedback,
  LoopShaping,
  ActuatorSaturation,
  IntegralWindup,
  AntiWindup,
  MeasurementNoise,
  SensorDynamics
 ]

 EXCLUDE_AS_PRIMARY = [
  OpenLoopControlWithoutFeedback,
  FeedforwardOnlyControl,
  PureStateEstimationWithoutControlAction,
  SignalFilteringWithoutClosedLoopSemantics
 ]

 DISAMBIGUATION_RULE =
  "When a controller uses measured or estimated output or state to update subsequent control action, interpret the context as feedback control."
}

CORE_CONCEPTS {

 FeedbackControl {
  definition = "Closed-loop control in which measured or estimated system behaviour influences future control action."
 }

 NegativeFeedback {
  definition = "Feedback in which the measured output is subtracted from the reference to form an error."
 }

 PositiveFeedback {
  definition = "Feedback in which returned system information reinforces rather than opposes the reference or current response."
 }

 ErrorSignal {
  definition = "The difference between a desired reference and measured output under standard negative-feedback convention."
 }

 LoopGain {
  definition = "The product of component transfer functions around the feedback loop."
 }

 ClosedLoopTransferFunction {
  definition = "The input-output transfer function after feedback interconnection."
 }

 SensitivityFunction {
  symbol = "S(s)"
  definition = "A function measuring how closed-loop behaviour responds to perturbations in loop dynamics."
 }

 ComplementarySensitivityFunction {
  symbol = "T(s)"
  definition = "A function describing reference transmission and measurement-noise pathways in standard feedback structures."
 }

 DisturbanceRejection {
  definition = "The attenuation of exogenous disturbances by the closed-loop system."
 }

 ReferenceTracking {
  definition = "The ability of the controlled output to follow a desired reference signal."
 }

 Regulation {
  definition = "The maintenance of desired system behaviour, commonly around a setpoint."
 }

 LoopShaping {
  definition = "The design of loop frequency response to achieve stability and performance objectives."
 }
}

RELATIONS {
 FeedbackControl --is_child_of--> ControlTheory
 FeedbackControl --uses--> FeedbackLoop
 FeedbackLoop --contains--> Controller
 FeedbackLoop --contains--> Plant
 FeedbackLoop --contains--> Sensor
 Controller --produces--> ControlInput
 Plant --produces--> ControlledOutput
 Sensor --measures--> ControlledOutput
 MeasuredOutput --is_compared_with--> ReferenceSignal
 ErrorSignal --drives--> Controller
 NegativeFeedback --reduces--> ErrorSignal
 LoopGain --determines--> ClosedLoopResponse
 LoopGain --influences--> StabilityMargins
 SensitivityFunction --relates_to--> DisturbanceSensitivity
 ComplementarySensitivityFunction --relates_to--> ReferenceTracking
 PIDControl --is_type_of--> FeedbackControl
 StateFeedback --is_type_of--> FeedbackControl
 OutputFeedback --is_type_of--> FeedbackControl
 ObserverBasedFeedback --uses--> StateEstimate
 AntiWindup --mitigates--> IntegralWindup
}

OPERATORS {
 ERROR_SIGNAL(r,y) :=
  e=r-y

 NEGATIVE_FEEDBACK_CONTROL(controller,e) :=
  u=controller(e)

 OPEN_LOOP_TRANSFER(C,G) :=
  L=C*G

 CLOSED_LOOP_TRANSFER(C,G,H) :=
  (C*G)/(1+C*G*H)
  under standard negative-feedback convention

 SENSITIVITY(L) :=
  S=1/(1+L)

 COMPLEMENTARY_SENSITIVITY(L) :=
  T=L/(1+L)

 UNITY_FEEDBACK_TRACKING(G,C) :=
  T=(G*C)/(1+G*C)

 DISTURBANCE_TO_OUTPUT(G,C) :=
  G/(1+G*C)
  for declared additive input-disturbance structure

 MEASUREMENT_NOISE_TO_OUTPUT(L) :=
  -T
  under standard unity-feedback measurement-noise convention

 PROPORTIONAL_CONTROL(e,Kp) :=
  Kp*e

 PI_CONTROL(e,Kp,Ki) :=
  Kp*e+Ki*integral(e dt)

 PD_CONTROL(e,Kp,Kd) :=
  Kp*e+Kd*de/dt

 PID_CONTROL(e,Kp,Ki,Kd) :=
  Kp*e+Ki*integral(e dt)+Kd*de/dt

 STATE_FEEDBACK(K,x) :=
  u=-K*x
}

DEFINITIONS {
 D1 = "Feedback control closes the loop between system behaviour and control action."
 D2 = "Negative feedback is the dominant convention for regulation and error correction."
 D3 = "Closed-loop performance depends on loop gain and feedback architecture."
 D4 = "The sensitivity function measures disturbance and model sensitivity."
 D5 = "The complementary sensitivity function measures reference transmission and some noise pathways."
 D6 = "Integral action reduces or removes steady-state error for suitable signal classes."
 D7 = "Derivative action adds predictive damping but can amplify measurement noise."
 D8 = "State feedback uses internal state or estimated state rather than only scalar error."
 D9 = "Actuator saturation can cause integral windup."
 D10 = "Robust feedback design balances performance against uncertainty and noise sensitivity."
}

AXIOMS {
 A1_CLOSED_LOOP:
  FeedbackControl contains a path from measured or estimated system behaviour back to control action

 A2_ERROR_CORRECTION:
  Negative feedback acts to reduce deviation between desired and measured behaviour under compatible controller design

 A3_STABILITY:
  Feedback is acceptable only when the resulting closed-loop system satisfies required stability criteria

 A4_LOOP_GAIN:
  Closed-loop sensitivity depends on loop gain

 A5_TRACKING_REJECTION_TRADEOFF:
  Improving tracking and disturbance rejection can increase control effort or noise sensitivity

 A6_SENSOR_DEPENDENCE:
  Feedback performance depends on measurement quality and sensor dynamics

 A7_ACTUATOR_LIMITS:
  Closed-loop design must respect actuator magnitude and rate limits

 A8_ROBUSTNESS:
  Feedback design should tolerate declared model uncertainty and disturbances
}

REASONING {

 R1_LOOP_EXTRACTION {
  INPUT = control_system_description
  IDENTIFY = reference_controller_plant_sensor_output_and_return path
  RETURN = feedback_loop_structure
 }

 R2_FEEDBACK_SIGN_ANALYSIS {
  INPUT = interconnection
  DETERMINE = negative_or_positive feedback
  RETURN = feedback_sign
 }

 R3_LOOP_TRANSFER_ANALYSIS {
  INPUT = controller_plant_sensor transfer models
  COMPUTE = open_loop_transfer
  COMPUTE = closed_loop_transfer
  RETURN = loop_transfer_profile
 }

 R4_STABILITY_ANALYSIS {
  INPUT = loop_transfer
  ANALYSE = poles_Nyquist_Bode_margins_or_root_locus
  RETURN = closed_loop_stability_profile
 }

 R5_TRACKING_REASONING {
  INPUT = reference_model_and_closed_loop
  COMPUTE = tracking_error
  ASSESS = steady_state_and_transient tracking
  RETURN = tracking_profile
 }

 R6_DISTURBANCE_REJECTION {
  INPUT = disturbance_location_and_loop model
  DERIVE = disturbance_to_output transfer
  ASSESS = attenuation
  RETURN = disturbance_rejection_profile
 }

 R7_NOISE_SENSITIVITY {
  INPUT = sensor_noise_pathway
  DERIVE = noise_transfer
  IDENTIFY = high_frequency amplification risk
  RETURN = noise_profile
 }

 R8_PID_REASONING {
  INPUT = tracking_regulation requirements
  SELECT = P_PI_PD_or_PID structure
  TUNE = gains
  CHECK = stability_transient_response_and_saturation
  RETURN = PID_profile
 }

 R9_STATE_OR_OUTPUT_FEEDBACK {
  INPUT = state_space_model_and_measurements
  IF state_available
   DESIGN = StateFeedback
  ELSE
   DESIGN = OutputFeedback_or_ObserverBasedFeedback
  RETURN = feedback_controller
 }

 R10_ROBUSTNESS_VALIDATION {
  INPUT = closed_loop_model_and_uncertainty
  CHECK = gain_phase margins
  CHECK = sensitivity peaks
  CHECK = actuator limits
  CHECK = model perturbations
  RETURN = robust_feedback_status
 }
}

CONSTRAINTS {
 C1 = "Feedback sign conventions must be explicit."
 C2 = "Closed-loop transfer formulas depend on block-diagram structure."
 C3 = "High loop gain can improve low-frequency disturbance rejection while worsening high-frequency noise sensitivity."
 C4 = "Derivative control requires noise-aware implementation."
 C5 = "Integral control must account for actuator saturation."
 C6 = "Gain and phase margins apply to compatible loop-transfer settings."
 C7 = "Positive feedback can destabilise systems if not deliberately constrained."
 C8 = "State feedback requires measurable or estimable state."
 C9 = "Sensor and actuator dynamics should not be omitted when they materially affect stability."
 C10 = "Nominal stability does not guarantee robust stability."
}

SEMANTIC_EQUIVALENCE {
 "FeedbackControl" <=> "ClosedLoopControl"
 "NegativeFeedback" <=> "ErrorReducingFeedback"
 "LoopGain" <=> "OpenLoopLoopTransfer"
 "SensitivityFunction" <=> "ClosedLoopSensitivity"
 "ComplementarySensitivityFunction" <=> "ClosedLoopComplementarySensitivity"
 "ReferenceTracking" <=> "SetpointFollowing"
}

VALIDATION {
 V1 = "Feedback loop structure is identified."
 V2 = "Feedback sign is explicit."
 V3 = "Reference, output and error signals are consistent."
 V4 = "Controller, plant and sensor dynamics are represented."
 V5 = "Closed-loop stability is checked."
 V6 = "Tracking and regulation metrics are evaluated."
 V7 = "Disturbance and noise pathways are analysed."
 V8 = "Actuator saturation is considered."
 V9 = "Robustness margins or equivalent evidence are documented."
 V10 = "Nominal and robust closed-loop status are distinguished."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 FEEDBACK_CONTROL_DEFINITION = 0.99
 LOOP_EXTRACTION_REASONING = 0.99
 LOOP_TRANSFER_REASONING = 0.99
 STABILITY_REASONING = 0.99
 TRACKING_REASONING = 0.99
 DISTURBANCE_REJECTION_REASONING = 0.98
 NOISE_REASONING = 0.98
 PID_REASONING = 0.98
 ROBUSTNESS_REASONING = 0.98

 POLICY =
  "Reduce confidence when block-diagram structure, feedback sign, plant dynamics, sensor dynamics, actuator limits, uncertainty or disturbance and noise pathways are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  FeedbackControlReasoning,
  ClosedLoopReasoning,
  LoopGainReasoning,
  SensitivityReasoning,
  ComplementarySensitivityReasoning,
  TrackingReasoning,
  RegulationReasoning,
  DisturbanceRejectionReasoning,
  NoiseSensitivityReasoning,
  PIDReasoning,
  StateFeedbackReasoning,
  OutputFeedbackReasoning,
  ObserverBasedFeedbackReasoning,
  LoopShapingReasoning,
  RobustFeedbackReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "feedback control" -> FeedbackControl
  "closed-loop control" -> FeedbackControl
  "negative feedback" -> NegativeFeedback
  "positive feedback" -> PositiveFeedback
  "loop gain" -> LoopGain
  "sensitivity function" -> SensitivityFunction
  "complementary sensitivity" -> ComplementarySensitivityFunction
  "reference tracking" -> ReferenceTracking
  "disturbance rejection" -> DisturbanceRejection
  "output feedback" -> OutputFeedback
 }

 INTERPRETATION_ORDER = [
  detect_feedback_control_context,
  identify reference_controller_plant_sensor_measurement_output_disturbance_noise_and_actuator pathways,
  determine feedback sign_and_loop architecture,
  construct loop_transfer_closed_loop_sensitivity_and_complementary_sensitivity models,
  analyse stability_and_margins,
  assess tracking_regulation_disturbance rejection_and_noise sensitivity,
  select PID_stateFeedback_outputFeedback_observerBased_or_loopShaping method,
  verify actuator_limits_integral_windup_and_robustness,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("feedback control")
  OR measured_or_estimated_output_affects_future_control
  THEN activate(FeedbackControlOntology)

 E2:
  IF standard_negative_unity_feedback
  THEN use S=1/(1+L) and T=L/(1+L)

 E3:
  IF integral_action_present
  THEN assess steady_state error_and_windup

 E4:
  IF derivative_action_present
  THEN assess measurement_noise sensitivity

 E5:
  IF state_available
  THEN consider state_feedback

 E6:
  IF state_not_directly_measured
  THEN consider observer_based feedback

 E7:
  IF frequency_domain_design
  THEN assess gain_margin_phase_margin_and_loop shape

 E8:
  IF model_uncertainty_material
  THEN assess robust feedback properties

 E9:
  IF actuator_saturation_occurs
  THEN include anti_windup_or_constraint handling
}

RECURSION {
 ENTRY = FeedbackControl

 EXPAND(FeedbackControl) -> [
  FeedbackLoop,
  NegativeFeedback,
  PositiveFeedback,
  ReferenceSignal,
  MeasuredOutput,
  ControlledOutput,
  ErrorSignal,
  Controller,
  Plant,
  Sensor,
  Actuator,
  ControlInput,
  LoopGain,
  OpenLoopTransferFunction,
  ClosedLoopTransferFunction,
  SensitivityFunction,
  ComplementarySensitivityFunction,
  DisturbanceRejection,
  NoiseRejection,
  ReferenceTracking,
  Regulation,
  Stability,
  ClosedLoopStability,
  GainMargin,
  PhaseMargin,
  BodePlot,
  NyquistCriterion,
  RootLocus,
  ProportionalControl,
  IntegralControl,
  DerivativeControl,
  PIDControl,
  StateFeedback,
  OutputFeedback,
  ObserverBasedFeedback,
  FeedforwardCompensation,
  RobustFeedback,
  LoopShaping,
  ActuatorSaturation,
  IntegralWindup,
  AntiWindup,
  MeasurementNoise,
  SensorDynamics
 ]

 EXPAND(ClosedLoopFunctions) -> [
  LoopGain,
  ClosedLoopTransferFunction,
  SensitivityFunction,
  ComplementarySensitivityFunction,
  DisturbanceTransferFunction,
  NoiseTransferFunction
 ]

 EXPAND(ControllerFamilies) -> [
  ProportionalControl,
  PIControl,
  PDControl,
  PIDControl,
  StateFeedback,
  OutputFeedback,
  ObserverBasedFeedback,
  RobustFeedback
 ]

 EXPAND(PerformanceMeasures) -> [
  TrackingError,
  SettlingTime,
  Overshoot,
  SteadyStateError,
  DisturbanceAttenuation,
  NoiseAmplification,
  GainMargin,
  PhaseMargin,
  ControlEffort
 ]

 STOP_CONDITION =
  "Stop when the loop structure, feedback sign, controller, plant, sensor, closed-loop stability, sensitivity, tracking, disturbance rejection, noise behaviour, actuator limits and robustness are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Control Theory"

 IMMEDIATE_CONTEXTS = [
  "Control Theory Ontology",
  "Stability Ontologies",
  "Frequency Response Ontologies",
  "PID Control Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Control Theory Ontology",
  "Control Theory",
  "Dynamical Systems Ontologies",
  "Linear Algebra Ontologies",
  "Signal Processing Ontologies",
  "State Space Ontologies",
  "PID Control Ontologies",
  "Robust Control Ontologies",
  "Observer Ontologies",
  "Kalman Filter Ontologies",
  "Model Predictive Control Ontologies",
  "Numerical Analysis Ontologies",
  "System Identification Ontologies"
 ]

 ROUTE_RULES = [
  "Feedback Control -> Control Theory Ontology for core control semantics",
  "Feedback Control -> Dynamical Systems for closed-loop trajectories and stability",
  "Feedback Control -> Signal Processing for frequency-domain and noise pathways",
  "Feedback Control -> State Space Ontologies for state feedback",
  "Feedback Control -> PID Control for classical error-based controller structure",
  "Feedback Control -> Robust Control for uncertainty and sensitivity shaping",
  "Feedback Control -> Observer and Kalman Filter Ontologies for estimated-state feedback",
  "Feedback Control -> Model Predictive Control for constrained feedback optimisation",
  "Feedback Control -> Numerical Analysis for simulation and discretisation",
  "Feedback Control -> System Identification for plant-model estimation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible feedback architectures, controllers or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve closed-loop semantics.",
  "Preserve feedback-sign semantics.",
  "Preserve error-signal semantics.",
  "Preserve sensitivity and complementary-sensitivity semantics.",
  "Preserve tracking-versus-disturbance-rejection distinctions.",
  "Preserve actuator and sensor limitations.",
  "Preserve nominal-versus-robust stability distinction.",
  "Preserve machine-readable feedback-control semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Feedback control uses measured or estimated system behaviour to modify future control action and achieve desired closed-loop behaviour."

 ERROR_SUMMARY =
  "In standard negative feedback, the error is the reference minus the measured output."

 LOOP_SUMMARY =
  "The loop gain combines controller, plant and feedback-path dynamics."

 SENSITIVITY_SUMMARY =
  "The sensitivity function describes how disturbances and model variations affect the closed loop."

 COMPLEMENTARY_SUMMARY =
  "The complementary sensitivity function describes reference transmission and important measurement-noise pathways."

 TRACKING_SUMMARY =
  "Reference tracking measures how closely the controlled output follows the desired signal."

 DISTURBANCE_SUMMARY =
  "Disturbance rejection measures how effectively feedback attenuates external perturbations."

 PID_SUMMARY =
  "PID control combines proportional, integral and derivative actions in an error-driven feedback law."

 ROBUSTNESS_SUMMARY =
  "Robust feedback design balances tracking, disturbance rejection, noise sensitivity, stability margins and actuator limits."
}

FLOW {
 INPUT
 -> detect_feedback_control_context
 -> identify reference_controller_plant_sensor_output_measurement_disturbance_noise_and_actuator structure
 -> determine negative_or_positive feedback sign
 -> construct open_loop_and_closed_loop models
 -> derive loop_gain_sensitivity_and_complementary_sensitivity
 -> analyse closed_loop poles_frequency response_gain margin_phase margin_or_root locus
 -> evaluate tracking_regulation_disturbance rejection_noise sensitivity_and_control effort
 -> select proportional_PI_PD_PID_state_feedback_output_feedback_observer_based_or_robust feedback method
 -> check actuator_saturation_integral_windup_sensor dynamics_and_model uncertainty
 -> route to control_theory_ontology dynamical_systems signal_processing state_space PID robust_control observer Kalman_filter MPC numerical_analysis_or_system_identification ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Feedback Control Ontology defines closed-loop regulation using measured or estimated system behaviour. It supports negative and positive feedback, reference, output and error signals, loop gain, open- and closed-loop transfer functions, sensitivity and complementary sensitivity, tracking, regulation, disturbance and noise rejection, gain and phase margins, Bode, Nyquist and root-locus analysis, proportional, PI, PD and PID control, state and output feedback, observer-based feedback, loop shaping, robustness, actuator saturation, integral windup and anti-windup reasoning."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-FEEDBACK-CONTROL-ONTOLOGY-V1"
}

}

END_FLOW