Adaptive Control Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Adaptive Control Ontology within the Kireth Research Semantic Library. Adaptive control studies controllers that modify their parameters or internal models online in response to changing system dynamics, uncertainty or operating conditions. This ontology supports direct and indirect adaptive control, model reference adaptive control, self-tuning regulators, gain scheduling, online parameter estimation, recursive least squares, adaptation laws, persistent excitation, stability, robustness, projection and dead-zone modifications, adaptive observers, certainty equivalence and relationships with control theory, system identification, estimation, nonlinear control and optimisation.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = AdaptiveControl
 SYMBOLIC_FORMS = [
  "u(t)=C(theta_hat(t),x(t),r(t))",
  "theta_hat_dot = adaptation_law(...)",
  "e_m(t)=y(t)-y_m(t)"
 ]
 ENTITY_CLASS = OnlineControllerAdjustmentFramework
 PRIMARY_CONTEXT = ControlTheory
 DEFINITION = "Adaptive control is control in which controller parameters, plant parameters or internal models are updated online in response to observed system behaviour."
 SEMANTIC_ROLE = "Provides the machine-readable framework for online parameter adaptation, model-reference control, self-tuning regulation, recursive estimation, excitation conditions, adaptive stability and robustness modifications."
}

SCOPE {
 INCLUDE = [
  AdaptiveControl,
  AdaptiveController,
  AdaptationLaw,
  DirectAdaptiveControl,
  IndirectAdaptiveControl,
  ModelReferenceAdaptiveControl,
  MRAC,
  SelfTuningRegulator,
  STR,
  GainScheduling,
  OnlineParameterEstimation,
  ParameterEstimate,
  ParameterError,
  RecursiveLeastSquares,
  GradientAdaptation,
  LyapunovAdaptation,
  CertaintyEquivalence,
  ReferenceModel,
  TrackingError,
  ModelError,
  PersistentExcitation,
  SufficientExcitation,
  ParameterConvergence,
  Boundedness,
  Stability,
  LyapunovStability,
  RobustAdaptiveControl,
  ProjectionOperator,
  DeadZoneModification,
  SigmaModification,
  EModification,
  Normalisation,
  AdaptiveObserver,
  UnknownParameter,
  TimeVaryingParameter,
  PlantUncertainty,
  ControllerGainUpdate,
  IdentificationModel,
  RegressorVector,
  AdaptationGain,
  ForgettingFactor
 ]

 EXCLUDE_AS_PRIMARY = [
  FixedGainControlWithoutOnlineAdaptation,
  GainSchedulingWithoutOperatingConditionMapping,
  PureSystemIdentificationWithoutClosedLoopControl,
  MachineLearningAdaptationWithoutControlSystemSemantics
 ]

 DISAMBIGUATION_RULE =
  "When controller or model parameters are adjusted online using measured system behaviour, interpret the context as adaptive control."
}

CORE_CONCEPTS {

 AdaptiveControl {
  definition = "Control with online modification of controller or model parameters."
 }

 AdaptationLaw {
  definition = "A rule governing how estimated parameters or controller gains evolve over time."
 }

 DirectAdaptiveControl {
  definition = "Adaptive control in which controller parameters are updated directly from performance signals."
 }

 IndirectAdaptiveControl {
  definition = "Adaptive control in which plant parameters are estimated online and then used to redesign or update the controller."
 }

 ModelReferenceAdaptiveControl {
  definition = "Adaptive control that adjusts controller parameters so plant behaviour tracks a prescribed reference model."
 }

 SelfTuningRegulator {
  definition = "An indirect adaptive controller combining online model estimation with controller redesign."
 }

 PersistentExcitation {
  definition = "An excitation condition ensuring regressors contain sufficient information for parameter convergence."
 }

 RecursiveLeastSquares {
  definition = "An online parameter-estimation method updating estimates recursively as new data arrive."
 }

 CertaintyEquivalence {
  definition = "The principle of using estimated parameters in a controller as though they were true parameters."
 }

 RobustAdaptiveControl {
  definition = "Adaptive control augmented to tolerate disturbances, noise, unmodelled dynamics or parameter drift."
 }
}

RELATIONS {
 AdaptiveControl --is_child_of--> ControlTheory
 AdaptiveController --uses--> AdaptationLaw
 DirectAdaptiveControl --updates--> ControllerParameters
 IndirectAdaptiveControl --updates--> PlantParameterEstimates
 IndirectAdaptiveControl --redesigns--> Controller
 ModelReferenceAdaptiveControl --uses--> ReferenceModel
 ModelReferenceAdaptiveControl --minimises--> TrackingError
 SelfTuningRegulator --uses--> OnlineParameterEstimation
 RecursiveLeastSquares --estimates--> UnknownParameter
 PersistentExcitation --supports--> ParameterConvergence
 CertaintyEquivalence --uses--> ParameterEstimate
 ProjectionOperator --bounds--> ParameterEstimate
 DeadZoneModification --reduces--> NoiseDrivenAdaptation
 SigmaModification --improves--> Robustness
 EModification --improves--> Robustness
 AdaptiveObserver --estimates--> StateAndOrParameters
 GainScheduling --maps--> OperatingConditionToControllerGain
}

OPERATORS {
 TRACKING_ERROR(y,y_m) :=
  e=y-y_m

 PARAMETER_ERROR(theta_hat,theta_true) :=
  theta_tilde=theta_hat-theta_true

 GRADIENT_ADAPTATION(theta_hat,phi,e,gamma) :=
  theta_hat_dot=-gamma*phi*e

 NORMALISED_GRADIENT_ADAPTATION(theta_hat,phi,e,gamma) :=
  theta_hat_dot=-(gamma*phi*e)/(1+phi^T*phi)

 RLS_GAIN_UPDATE(P,phi,lambda) :=
  K=P*phi/(lambda+phi^T*P*phi)

 RLS_PARAMETER_UPDATE(theta_hat,K,error) :=
  theta_hat_new=theta_hat+K*error

 RLS_COVARIANCE_UPDATE(P,K,phi,lambda) :=
  P_new=(P-K*phi^T*P)/lambda

 PROJECTION(theta_hat_dot,theta_hat,bounds) :=
  modified_update_that_keeps_theta_hat_within_bounds

 DEAD_ZONE(error,delta) :=
  0 if abs(error)<=delta
  else error

 SIGMA_MODIFICATION(theta_hat_dot_nominal,sigma,theta_hat) :=
  theta_hat_dot_nominal-sigma*theta_hat

 CERTAINTY_EQUIVALENT_CONTROL(theta_hat,state,reference) :=
  control_law_using_current_parameter_estimate
}

DEFINITIONS {
 D1 = "Adaptive control modifies controller or model parameters online."
 D2 = "Direct adaptive control updates controller parameters without explicitly identifying the plant first."
 D3 = "Indirect adaptive control estimates plant parameters and then updates the controller."
 D4 = "MRAC attempts to make plant behaviour follow a reference model."
 D5 = "Self-tuning regulators combine recursive identification with controller redesign."
 D6 = "Persistent excitation supports parameter convergence."
 D7 = "Tracking convergence does not always imply parameter convergence."
 D8 = "Noise and unmodelled dynamics can destabilise naive adaptation."
 D9 = "Projection, dead-zone and sigma modifications improve robustness."
 D10 = "Certainty equivalence treats estimated parameters as current best model values."
}

AXIOMS {
 A1_ONLINE_ADAPTATION:
  AdaptiveControl updates one or more parameters during operation

 A2_MEASUREMENT_DEPENDENCE:
  Adaptation depends on measured or estimated system behaviour

 A3_MODEL_OR_PERFORMANCE_REFERENCE:
  Adaptive control requires a model, error signal or performance criterion that drives adaptation

 A4_STABILITY:
  Adaptation law design should preserve bounded closed-loop behaviour under declared assumptions

 A5_EXCITATION:
  Parameter convergence generally requires sufficient excitation

 A6_ESTIMATION_CONTROL_COUPLING:
  In indirect adaptive control, estimator behaviour influences controller behaviour

 A7_ROBUSTNESS:
  Disturbance and noise sensitivity must be accounted for in practical adaptive laws

 A8_PARAMETER_BOUNDS:
  Known physical or safe parameter bounds should be enforced when relevant
}

REASONING {

 R1_ADAPTIVE_CONTEXT_IDENTIFICATION {
  INPUT = control_problem
  IDENTIFY = unknown_or_varying parameters
  IDENTIFY = measurable signals
  IDENTIFY = controller parameters_or_model parameters
  RETURN = adaptive_control_context
 }

 R2_ARCHITECTURE_CLASSIFICATION {
  INPUT = adaptive_control_context
  CLASSIFY = direct_indirect_MRAC_selfTuning_gainScheduling_or_adaptiveObserver
  RETURN = adaptive_architecture
 }

 R3_REFERENCE_MODEL_REASONING {
  INPUT = plant_output_reference_model_output
  COMPUTE = tracking_error
  DESIGN = adaptation law_to_reduce error
  RETURN = MRAC_structure
 }

 R4_ONLINE_ESTIMATION {
  INPUT = regressor_measurements_and_parameter model
  SELECT = RLS_gradient_or_other estimator
  UPDATE = parameter_estimates
  RETURN = estimated_model
 }

 R5_CONTROLLER_UPDATE {
  INPUT = current_parameter_estimates
  APPLY = certainty_equivalence_or_direct gain update
  RETURN = updated_controller
 }

 R6_EXCITATION_ANALYSIS {
  INPUT = regressor_history
  TEST = persistent_or_sufficient excitation
  RETURN = parameter_convergence potential
 }

 R7_LYAPUNOV_ADAPTATION_REASONING {
  INPUT = candidate_Lyapunov_function
  DERIVE = adaptation law
  CHECK = V_dot_nonpositive
  RETURN = boundedness_or_stability argument
 }

 R8_ROBUSTNESS_MODIFICATION {
  INPUT = noise_unmodelled_dynamics_or_disturbance
  SELECT = projection_deadZone_sigmaModification_eModification_or_normalisation
  RETURN = robustified_adaptation law
 }

 R9_TIME_VARYING_PARAMETER_REASONING {
  INPUT = drifting_parameters
  SELECT = forgetting_factor_tracking estimator_or_adaptation gain
  RETURN = parameter_tracking profile
 }

 R10_ADAPTIVE_VALIDATION {
  INPUT = adaptive_closed_loop
  CHECK = boundedness_tracking_parameter estimates_control effort_and_noise sensitivity
  RETURN = adaptive_control_status
 }
}

CONSTRAINTS {
 C1 = "Parameterisation must be compatible with the adaptive law."
 C2 = "Persistent excitation conditions must not be assumed without evidence."
 C3 = "Tracking convergence does not guarantee exact parameter identification."
 C4 = "Noise can produce parameter drift."
 C5 = "High adaptation gains can amplify noise or excite unmodelled dynamics."
 C6 = "Forgetting factors improve tracking of changing parameters but increase estimator variance."
 C7 = "Projection bounds must reflect physically or operationally valid parameter ranges."
 C8 = "Certainty-equivalent control can fail if estimates are poor or dynamics are insufficiently excited."
 C9 = "Gain scheduling is not identical to true adaptation unless gains respond online to operating conditions."
 C10 = "Stability guarantees depend on the declared plant and uncertainty assumptions."
}

SEMANTIC_EQUIVALENCE {
 "AdaptiveControl" <=> "OnlineAdaptiveFeedbackControl"
 "ModelReferenceAdaptiveControl" <=> "MRAC"
 "SelfTuningRegulator" <=> "STR"
 "RecursiveLeastSquares" <=> "RLS"
 "PersistentExcitation" <=> "InformativeExcitationCondition"
 "CertaintyEquivalence" <=> "ControlUsingEstimatedParametersAsTrue"
}

VALIDATION {
 V1 = "Unknown or varying parameters are identified."
 V2 = "Adaptive architecture is classified."
 V3 = "Adaptation signals are explicit."
 V4 = "Parameter estimator assumptions are documented."
 V5 = "Excitation conditions are assessed."
 V6 = "Closed-loop boundedness or stability is assessed."
 V7 = "Noise and disturbance sensitivity are considered."
 V8 = "Parameter bounds are enforced where required."
 V9 = "Tracking and parameter convergence are reported separately."
 V10 = "Nominal and robust adaptive-control claims are distinguished."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 ADAPTIVE_CONTROL_DEFINITION = 0.99
 ARCHITECTURE_REASONING = 0.99
 MRAC_REASONING = 0.98
 ONLINE_ESTIMATION_REASONING = 0.98
 EXCITATION_REASONING = 0.97
 LYAPUNOV_ADAPTATION_REASONING = 0.98
 ROBUSTIFICATION_REASONING = 0.98
 TIME_VARYING_REASONING = 0.97
 VALIDATION_REASONING = 0.99

 POLICY =
  "Reduce confidence when parameterisation, excitation, estimator noise, adaptation gains, uncertainty bounds, reference model or robustness assumptions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  AdaptiveControlReasoning,
  DirectAdaptiveControlReasoning,
  IndirectAdaptiveControlReasoning,
  MRACReasoning,
  SelfTuningRegulatorReasoning,
  RecursiveLeastSquaresReasoning,
  GradientAdaptationReasoning,
  PersistentExcitationReasoning,
  ParameterConvergenceReasoning,
  CertaintyEquivalenceReasoning,
  ProjectionReasoning,
  DeadZoneReasoning,
  SigmaModificationReasoning,
  AdaptiveObserverReasoning,
  RobustAdaptiveControlReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "adaptive control" -> AdaptiveControl
  "adaptive controller" -> AdaptiveController
  "model reference adaptive control" -> ModelReferenceAdaptiveControl
  "MRAC" -> ModelReferenceAdaptiveControl
  "self-tuning regulator" -> SelfTuningRegulator
  "STR" -> SelfTuningRegulator
  "recursive least squares" -> RecursiveLeastSquares
  "RLS" -> RecursiveLeastSquares
  "persistent excitation" -> PersistentExcitation
  "certainty equivalence" -> CertaintyEquivalence
 }

 INTERPRETATION_ORDER = [
  detect_adaptive_control_context,
  identify unknown_or_timeVarying parameters_measurements_reference_model_and_controller structure,
  classify direct_indirect_MRAC_STR_gainScheduling_or_adaptiveObserver architecture,
  select gradient_RLS_Lyapunov_or_other adaptation law,
  update parameter estimates_or_controller gains,
  assess excitation_parameter convergence_tracking_and_boundedness,
  apply projection_deadZone_sigma_eModification_or_normalisation if required,
  verify robustness_noise sensitivity_and_control effort,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("adaptive control")
  OR controller_parameters_are_updated_online
  THEN activate(AdaptiveControlOntology)

 E2:
  IF controller_parameters_updated_directly_from_tracking_error
  THEN classify DirectAdaptiveControl

 E3:
  IF plant_parameters_estimated_then_controller redesigned
  THEN classify IndirectAdaptiveControl

 E4:
  IF plant_output_tracks_reference_model
  THEN classify ModelReferenceAdaptiveControl

 E5:
  IF recursive_model_estimation_and_controller retuning
  THEN classify SelfTuningRegulator

 E6:
  IF parameter_convergence claimed
  THEN verify persistent_or_sufficient excitation

 E7:
  IF noisy_measurements
  THEN consider deadZone_projection_normalisation_or_robust modification

 E8:
  IF parameters_drift_over_time
  THEN consider forgetting_factor_or_tracking adaptation

 E9:
  IF adaptation_causes_unbounded growth
  THEN stop_or_modify adaptation_and_report instability risk
}

RECURSION {
 ENTRY = AdaptiveControl

 EXPAND(AdaptiveControl) -> [
  AdaptiveController,
  AdaptationLaw,
  DirectAdaptiveControl,
  IndirectAdaptiveControl,
  ModelReferenceAdaptiveControl,
  MRAC,
  SelfTuningRegulator,
  STR,
  GainScheduling,
  OnlineParameterEstimation,
  ParameterEstimate,
  ParameterError,
  RecursiveLeastSquares,
  GradientAdaptation,
  LyapunovAdaptation,
  CertaintyEquivalence,
  ReferenceModel,
  TrackingError,
  ModelError,
  PersistentExcitation,
  SufficientExcitation,
  ParameterConvergence,
  Boundedness,
  Stability,
  LyapunovStability,
  RobustAdaptiveControl,
  ProjectionOperator,
  DeadZoneModification,
  SigmaModification,
  EModification,
  Normalisation,
  AdaptiveObserver,
  UnknownParameter,
  TimeVaryingParameter,
  PlantUncertainty,
  ControllerGainUpdate,
  IdentificationModel,
  RegressorVector,
  AdaptationGain,
  ForgettingFactor
 ]

 EXPAND(AdaptiveArchitectures) -> [
  DirectAdaptiveControl,
  IndirectAdaptiveControl,
  ModelReferenceAdaptiveControl,
  SelfTuningRegulator,
  GainScheduling,
  AdaptiveObserver
 ]

 EXPAND(AdaptationMethods) -> [
  GradientAdaptation,
  LyapunovAdaptation,
  RecursiveLeastSquares,
  ProjectionOperator,
  DeadZoneModification,
  SigmaModification,
  EModification,
  Normalisation
 ]

 EXPAND(ConvergenceConditions) -> [
  Boundedness,
  Stability,
  PersistentExcitation,
  ParameterConvergence,
  TrackingConvergence
 ]

 STOP_CONDITION =
  "Stop when the adaptive architecture, unknown parameters, estimator or adaptation law, excitation, tracking, parameter convergence, robustness modification and closed-loop status are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Control Theory"

 IMMEDIATE_CONTEXTS = [
  "Control Theory Ontology",
  "Feedback Control Ontology",
  "System Identification Ontologies",
  "Estimation Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Control Theory Ontology",
  "Control Theory",
  "Feedback Control Ontology",
  "Control System Ontology",
  "State-Space Control Ontology",
  "System Identification Ontologies",
  "Parameter Estimation Ontology",
  "Recursive Least Squares Ontologies",
  "Nonlinear Control Ontologies",
  "Robust Control Ontologies",
  "Optimal Control Ontologies",
  "Signal Processing Ontologies",
  "Probability & Statistics Ontologies"
 ]

 ROUTE_RULES = [
  "Adaptive Control -> Control Theory Ontology for core control semantics",
  "Adaptive Control -> Feedback Control for closed-loop adaptation",
  "Adaptive Control -> Control System for architecture and signals",
  "Adaptive Control -> State-Space Control for state-model adaptive design",
  "Adaptive Control -> System Identification for online plant estimation",
  "Adaptive Control -> Parameter Estimation for recursive parameter inference",
  "Adaptive Control -> Nonlinear Control for Lyapunov-based adaptation",
  "Adaptive Control -> Robust Control for disturbance and unmodelled-dynamics protection",
  "Adaptive Control -> Signal Processing for filtering and regressor processing",
  "Adaptive Control -> Probability & Statistics for stochastic estimation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

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

 RULES = [
  "Preserve online-adaptation semantics.",
  "Preserve direct-versus-indirect adaptive distinction.",
  "Preserve MRAC and self-tuning regulator distinctions.",
  "Preserve tracking-versus-parameter-convergence distinction.",
  "Preserve excitation semantics.",
  "Preserve robust-modification semantics.",
  "Preserve parameter-bound semantics.",
  "Preserve machine-readable adaptive-control semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Adaptive control modifies controller or model parameters online in response to observed system behaviour and changing uncertainty."

 DIRECT_SUMMARY =
  "Direct adaptive control updates controller parameters directly from performance signals."

 INDIRECT_SUMMARY =
  "Indirect adaptive control estimates plant parameters and then redesigns or updates the controller."

 MRAC_SUMMARY =
  "Model reference adaptive control adjusts parameters so plant behaviour follows a reference model."

 STR_SUMMARY =
  "A self-tuning regulator combines online identification with controller redesign."

 EXCITATION_SUMMARY =
  "Persistent excitation provides the information richness needed for many parameter-convergence guarantees."

 ROBUSTNESS_SUMMARY =
  "Projection, dead-zone, sigma and related modifications improve robustness to noise, disturbance and unmodelled dynamics."

 ESTIMATION_SUMMARY =
  "Recursive least squares and gradient adaptation are common online parameter-estimation methods."

 STATUS_SUMMARY =
  "Adaptive-control results should distinguish bounded tracking, parameter convergence, robustness and identification accuracy."
}

FLOW {
 INPUT
 -> detect_adaptive_control_context
 -> identify plant_unknown_or_drifting parameters_measurements_reference signals_reference model_controller gains_and_uncertainty
 -> classify direct_indirect_MRAC_selfTuning_gainScheduling_or_adaptiveObserver architecture
 -> choose gradient_RLS_Lyapunov_or_other adaptation mechanism
 -> update controller_parameters_or_plant parameter estimates_online
 -> if_indirect redesign_controller_using current estimates
 -> analyse tracking_error_model error_boundedness_stability_and_parameter convergence
 -> assess persistent_excitation_or_information richness
 -> apply projection_deadZone_sigma_eModification_normalisation_or_forgetting factor as required
 -> verify noise sensitivity_control effort_parameter bounds_and_robustness
 -> route to control_theory_ontology feedback_control control_system state_space system_identification parameter_estimation nonlinear_control robust_control signal_processing_or_probability_statistics ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Adaptive Control Ontology defines controllers and models that update online in response to uncertain or changing dynamics. It supports direct and indirect adaptive control, model reference adaptive control, self-tuning regulators, gain scheduling, recursive least squares, gradient and Lyapunov adaptation, certainty equivalence, parameter errors, persistent excitation, tracking and parameter convergence, adaptive observers, projection, dead-zone, sigma and e-modifications, normalisation, forgetting factors, robust adaptation and explicit closed-loop boundedness and robustness reasoning."
}

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

}

END_FLOW