Robust Control Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Robust Control Ontology within the Kireth Research Semantic Library. Robust control studies the design and analysis of controllers that maintain stability and acceptable performance despite model uncertainty, disturbances, parameter variation and unmodelled dynamics. This ontology supports uncertainty sets, structured and unstructured uncertainty, robust stability, robust performance, sensitivity and complementary sensitivity, small-gain reasoning, H-infinity control, mu-analysis, loop shaping, gain and phase margins, worst-case analysis, disturbance attenuation, model perturbations and relationships with control theory, feedback control, state-space control, optimisation and system identification.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = RobustControl
 SYMBOLIC_FORMS = [
  "||T_zw||_infinity < gamma",
  "S(s)=1/(1+L(s))",
  "T(s)=L(s)/(1+L(s))"
 ]
 ENTITY_CLASS = UncertaintyAwareControlFramework
 PRIMARY_CONTEXT = ControlTheory
 DEFINITION = "Robust control is the analysis and design of controllers that preserve stability and specified performance across a declared set of model uncertainties and disturbances."
 SEMANTIC_ROLE = "Provides the machine-readable framework for model uncertainty, robust stability, robust performance, H-infinity norms, structured singular value, loop shaping, worst-case disturbance attenuation and uncertainty-aware controller design."
}

SCOPE {
 INCLUDE = [
  RobustControl,
  RobustStability,
  RobustPerformance,
  ModelUncertainty,
  ParametricUncertainty,
  StructuredUncertainty,
  UnstructuredUncertainty,
  AdditiveUncertainty,
  MultiplicativeUncertainty,
  InputMultiplicativeUncertainty,
  OutputMultiplicativeUncertainty,
  UnmodelledDynamics,
  Disturbance,
  Noise,
  SensitivityFunction,
  ComplementarySensitivityFunction,
  ControlSensitivity,
  LoopGain,
  LoopShaping,
  SmallGainTheorem,
  HInfinityNorm,
  HInfinityControl,
  HInfinitySynthesis,
  MixedSensitivity,
  StructuredSingularValue,
  MuAnalysis,
  MuSynthesis,
  DKIteration,
  GainMargin,
  PhaseMargin,
  DiskMargin,
  WorstCaseGain,
  WorstCasePerformance,
  NominalStability,
  NominalPerformance,
  UncertaintyWeight,
  PerformanceWeight,
  GeneralisedPlant,
  LowerLinearFractionalTransformation,
  UpperLinearFractionalTransformation,
  LFT,
  DisturbanceAttenuation,
  RobustPolePlacement,
  RobustStateFeedback,
  RobustObserver,
  RobustMPC,
  IntegralQuadraticConstraint,
  IQC
 ]

 EXCLUDE_AS_PRIMARY = [
  NominalControlWithoutUncertaintyModel,
  AdaptiveControlWhosePrimaryMechanismIsOnlineParameterUpdate,
  StochasticControlWhereUncertaintyIsPurelyProbabilisticWithoutWorstCaseSemantics,
  GenericSensitivityAnalysisWithoutControlContext
 ]

 DISAMBIGUATION_RULE =
  "When controller analysis or design explicitly requires stability or performance to hold across model uncertainty, perturbations or worst-case disturbances, interpret the context as robust control."
}

CORE_CONCEPTS {

 RobustControl {
  definition = "Control designed to maintain required behaviour despite uncertainty."
 }

 ModelUncertainty {
  definition = "A declared set of differences between a nominal model and possible true plants."
 }

 RobustStability {
  definition = "The property that the closed-loop system remains stable for all admissible uncertainties."
 }

 RobustPerformance {
  definition = "The property that declared performance requirements hold for all admissible uncertainties."
 }

 SensitivityFunction {
  symbol = "S"
  definition = "The closed-loop transfer function 1/(1+L) in a standard feedback loop."
 }

 ComplementarySensitivityFunction {
  symbol = "T"
  definition = "The closed-loop transfer function L/(1+L) in a standard feedback loop."
 }

 SmallGainTheorem {
  definition = "A sufficient stability result based on the product of induced gains being less than one."
 }

 HInfinityNorm {
  definition = "The maximum singular value of a stable transfer matrix over frequency."
 }

 StructuredSingularValue {
  symbol = "mu"
  definition = "A measure of robustness against structured uncertainty."
 }

 MixedSensitivity {
  definition = "A robust-control formulation shaping weighted sensitivity, control effort and complementary sensitivity simultaneously."
 }

 GeneralisedPlant {
  definition = "An augmented plant representation containing exogenous inputs, regulated outputs, measurements and control inputs for robust synthesis."
 }
}

RELATIONS {
 RobustControl --is_child_of--> ControlTheory
 RobustControl --uses--> ModelUncertainty
 RobustControl --requires--> RobustStability
 RobustPerformance --implies--> RobustStability
 ModelUncertainty --may_be--> ParametricUncertainty
 ModelUncertainty --may_be--> StructuredUncertainty
 ModelUncertainty --may_be--> UnstructuredUncertainty
 SensitivityFunction --relates_to--> DisturbanceRejection
 ComplementarySensitivityFunction --relates_to--> NoiseTransmission
 LoopShaping --modifies--> LoopGain
 SmallGainTheorem --supports--> RobustStability
 HInfinityControl --minimises--> WorstCaseGain
 MixedSensitivity --uses--> PerformanceWeight
 MixedSensitivity --uses--> UncertaintyWeight
 MuAnalysis --uses--> StructuredSingularValue
 MuSynthesis --may_use--> DKIteration
 GeneralisedPlant --supports--> HInfinitySynthesis
 LFT --represents--> InterconnectionWithUncertainty
}

OPERATORS {
 SENSITIVITY(L) :=
  1/(1+L)

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

 HINF_NORM(G) :=
  sup_omega sigma_max(G(j*omega))

 SMALL_GAIN_TEST(M,Delta) :=
  ||M||_infinity * ||Delta||_infinity < 1

 MULTIPLICATIVE_UNCERTAINTY_MODEL(G0,W,Delta) :=
  G=G0*(1+W*Delta)
  with ||Delta||_infinity<=1

 ADDITIVE_UNCERTAINTY_MODEL(G0,W,Delta) :=
  G=G0+W*Delta
  with ||Delta||_infinity<=1

 MIXED_SENSITIVITY_OBJECTIVE(Ws,S,Wu,K,Wt,T) :=
  || [Ws*S; Wu*K*S; Wt*T] ||_infinity

 ROBUST_STABILITY_MARGIN(mu_value) :=
  1/mu_value
  when mu_value>0

 LFT_LOWER(P,Delta) :=
  lower_linear_fractional_transformation

 LFT_UPPER(P,K) :=
  upper_linear_fractional_transformation

 WORST_CASE_GAIN(model_set,input_output_map) :=
  max_over_admissible_uncertainty induced_gain

 DISK_MARGIN(loop) :=
  multiloop_or_siso simultaneous gain_phase robustness measure
}

DEFINITIONS {
 D1 = "Robust control distinguishes nominal models from admissible true plants."
 D2 = "Robust stability requires stability for every plant in the uncertainty set."
 D3 = "Robust performance requires performance as well as stability across uncertainty."
 D4 = "Sensitivity and complementary sensitivity encode fundamental feedback trade-offs."
 D5 = "H-infinity control minimises worst-case induced gain."
 D6 = "Mu-analysis addresses structured uncertainty."
 D7 = "Loop shaping designs frequency-domain robustness and performance."
 D8 = "Uncertainty weights encode the frequency dependence of model error."
 D9 = "Small-gain reasoning provides a sufficient robust-stability condition."
 D10 = "Nominal success does not imply robust success."
}

AXIOMS {
 A1_UNCERTAINTY_SET:
  RobustControl requires an explicitly declared admissible uncertainty description

 A2_NOMINAL_MODEL:
  Robust analysis distinguishes a nominal plant from perturbations

 A3_ROBUST_STABILITY:
  ClosedLoopSystem must remain stable for all admissible uncertainties

 A4_ROBUST_PERFORMANCE:
  Performance constraints must hold for all admissible uncertainties when robust performance is claimed

 A5_SENSITIVITY_TRADEOFF:
  Sensitivity reduction in one frequency region creates trade-offs elsewhere under fundamental feedback limitations

 A6_WORST_CASE:
  Robust-control metrics are evaluated against worst-case admissible perturbations

 A7_WEIGHTING:
  Frequency weights must represent intended uncertainty or performance emphasis

 A8_CERTIFICATION:
  Robustness claims require mathematical or computational evidence tied to the declared uncertainty model
}

REASONING {

 R1_UNCERTAINTY_EXTRACTION {
  INPUT = control_problem
  IDENTIFY = nominal_plant
  IDENTIFY = parametric_uncertainty
  IDENTIFY = additive_or_multiplicative uncertainty
  IDENTIFY = disturbance_and_noise channels
  RETURN = uncertainty_model
 }

 R2_NOMINAL_ANALYSIS {
  INPUT = nominal_closed_loop
  CHECK = nominal_stability
  CHECK = nominal_performance
  RETURN = nominal_profile
 }

 R3_SENSITIVITY_REASONING {
  INPUT = loop_gain
  COMPUTE = S_and_T
  ASSESS = disturbance_rejection_noise_transmission_and_bandwidth
  RETURN = sensitivity_profile
 }

 R4_SMALL_GAIN_REASONING {
  INPUT = uncertainty_block_and_closed_loop transfer
  COMPUTE = induced_norm bounds
  IF product<1
   RETURN = robust_stability_sufficient
  ELSE
   RETURN = inconclusive_or_failed
 }

 R5_HINFINITY_REASONING {
  INPUT = generalised_plant
  FORM = regulated_and_measured channels
  SOLVE = HInfinity_synthesis
  COMPUTE = closed_loop_HInfinity norm
  RETURN = HInfinity_controller_and_gamma
 }

 R6_MIXED_SENSITIVITY_REASONING {
  INPUT = Ws_Wu_Wt
  FORM = weighted_sensitivity objective
  SOLVE = HInfinity_controller
  RETURN = shaped_robust_controller
 }

 R7_MU_ANALYSIS {
  INPUT = structured_uncertainty_interconnection
  COMPUTE = frequency_dependent mu upper_and_lower bounds
  RETURN = robust_stability_or_performance margins
 }

 R8_DK_ITERATION {
  INPUT = generalised_plant_and_uncertainty structure
  ALTERNATE = D_scaling_and_controller synthesis
  RETURN = mu_synthesis_controller
 }

 R9_WORST_CASE_ANALYSIS {
  INPUT = uncertain_model set
  SEARCH = worst_case uncertainty combination
  COMPUTE = worst_case_gain_or_performance
  RETURN = robust_margin_profile
 }

 R10_ROBUST_VALIDATION {
  INPUT = candidate_controller
  VERIFY = nominal_stability
  VERIFY = robust_stability
  VERIFY = nominal_performance
  VERIFY = robust_performance
  RETURN = robust_control_status
 }
}

CONSTRAINTS {
 C1 = "Robustness results are only as meaningful as the declared uncertainty model."
 C2 = "Uncertainty weights must be frequency-consistent with modelling error."
 C3 = "Small-gain conditions are sufficient but not always necessary."
 C4 = "H-infinity norms require stable transfer matrices for standard interpretations."
 C5 = "Mu upper and lower bounds may not coincide exactly."
 C6 = "Aggressive loop shaping can increase control effort or noise sensitivity."
 C7 = "High-frequency uncertainty should not be ignored in controller design."
 C8 = "Nominal gain and phase margins do not fully characterise structured multivariable robustness."
 C9 = "Actuator saturation can invalidate linear robust-performance predictions."
 C10 = "Simulation over a few uncertainty samples does not certify robust stability."
}

SEMANTIC_EQUIVALENCE {
 "RobustControl" <=> "UncertaintyAwareControl"
 "RobustStability" <=> "StabilityForAllAdmissibleModels"
 "RobustPerformance" <=> "PerformanceForAllAdmissibleModels"
 "HInfinityControl" <=> "WorstCaseInducedGainControl"
 "StructuredSingularValue" <=> "Mu"
 "LoopShaping" <=> "FrequencyDomainRobustnessShaping"
}

VALIDATION {
 V1 = "Nominal plant is identified."
 V2 = "Uncertainty structure is explicit."
 V3 = "Uncertainty bounds or weights are valid."
 V4 = "Nominal closed-loop stability is checked."
 V5 = "Robust stability is checked."
 V6 = "Performance channels and weights are explicit."
 V7 = "Sensitivity and complementary sensitivity are analysed."
 V8 = "Worst-case or mu evidence is documented."
 V9 = "Actuator and sensor limitations are considered."
 V10 = "Nominal and robust performance statuses are distinguished."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 ROBUST_CONTROL_DEFINITION = 0.99
 UNCERTAINTY_REASONING = 0.99
 SENSITIVITY_REASONING = 0.99
 SMALL_GAIN_REASONING = 0.98
 HINFINITY_REASONING = 0.98
 MIXED_SENSITIVITY_REASONING = 0.98
 MU_ANALYSIS_REASONING = 0.97
 WORST_CASE_REASONING = 0.97
 ROBUST_VALIDATION_REASONING = 0.99

 POLICY =
  "Reduce confidence when uncertainty structure, uncertainty weights, regulated outputs, disturbance channels, actuator limits, performance weights or worst-case evidence are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  RobustControlReasoning,
  RobustStabilityReasoning,
  RobustPerformanceReasoning,
  UncertaintyModelReasoning,
  SensitivityReasoning,
  ComplementarySensitivityReasoning,
  SmallGainReasoning,
  HInfinityReasoning,
  MixedSensitivityReasoning,
  MuAnalysisReasoning,
  MuSynthesisReasoning,
  LoopShapingReasoning,
  WorstCaseAnalysis,
  DisturbanceAttenuationReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "robust control" -> RobustControl
  "robust stability" -> RobustStability
  "robust performance" -> RobustPerformance
  "H-infinity control" -> HInfinityControl
  "H∞ control" -> HInfinityControl
  "mu analysis" -> MuAnalysis
  "structured singular value" -> StructuredSingularValue
  "loop shaping" -> LoopShaping
  "small gain theorem" -> SmallGainTheorem
  "mixed sensitivity" -> MixedSensitivity
 }

 INTERPRETATION_ORDER = [
  detect_robust_control_context,
  identify nominal_model_uncertainty_structure_disturbance_noise_channels_actuator_limits_and_performance requirements,
  classify parametric_structured_unstructured_additive_or_multiplicative uncertainty,
  construct sensitivity_complementarySensitivity_or_LFT representation,
  test nominal stability_and_performance,
  apply smallGain_HInfinity_mixedSensitivity_mu_loopShaping_or_worstCase pathway,
  derive robust margins_or_controller,
  verify robust_stability_and_robust_performance,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("robust control")
  OR requires_control_performance_across_model uncertainty
  THEN activate(RobustControlOntology)

 E2:
  IF uncertainty_is_unstructured_and_norm_bounded
  THEN consider small_gain_or_HInfinity reasoning

 E3:
  IF uncertainty_is_structured
  THEN consider mu_analysis

 E4:
  IF sensitivity_shaping_required
  THEN consider mixed_sensitivity formulation

 E5:
  IF performance_weights_defined
  THEN verify weight_meaning_and_frequency scaling

 E6:
  IF robust_stability_claimed
  THEN test entire_declared_uncertainty set

 E7:
  IF robust_performance_claimed
  THEN require both robust stability_and weighted performance conditions

 E8:
  IF controller_has_high_gain
  THEN assess noise_amplification_and_actuator demand

 E9:
  IF only_nominal_analysis_available
  THEN do_not claim robust stability
}

RECURSION {
 ENTRY = RobustControl

 EXPAND(RobustControl) -> [
  RobustStability,
  RobustPerformance,
  ModelUncertainty,
  ParametricUncertainty,
  StructuredUncertainty,
  UnstructuredUncertainty,
  AdditiveUncertainty,
  MultiplicativeUncertainty,
  InputMultiplicativeUncertainty,
  OutputMultiplicativeUncertainty,
  UnmodelledDynamics,
  Disturbance,
  Noise,
  SensitivityFunction,
  ComplementarySensitivityFunction,
  ControlSensitivity,
  LoopGain,
  LoopShaping,
  SmallGainTheorem,
  HInfinityNorm,
  HInfinityControl,
  HInfinitySynthesis,
  MixedSensitivity,
  StructuredSingularValue,
  MuAnalysis,
  MuSynthesis,
  DKIteration,
  GainMargin,
  PhaseMargin,
  DiskMargin,
  WorstCaseGain,
  WorstCasePerformance,
  NominalStability,
  NominalPerformance,
  UncertaintyWeight,
  PerformanceWeight,
  GeneralisedPlant,
  LowerLinearFractionalTransformation,
  UpperLinearFractionalTransformation,
  LFT,
  DisturbanceAttenuation,
  RobustPolePlacement,
  RobustStateFeedback,
  RobustObserver,
  RobustMPC,
  IntegralQuadraticConstraint,
  IQC
 ]

 EXPAND(UncertaintyTypes) -> [
  ParametricUncertainty,
  StructuredUncertainty,
  UnstructuredUncertainty,
  AdditiveUncertainty,
  InputMultiplicativeUncertainty,
  OutputMultiplicativeUncertainty
 ]

 EXPAND(RobustMethods) -> [
  SmallGainTheorem,
  HInfinityControl,
  MixedSensitivity,
  MuAnalysis,
  MuSynthesis,
  DKIteration,
  LoopShaping,
  IntegralQuadraticConstraint
 ]

 EXPAND(RobustMetrics) -> [
  RobustStabilityMargin,
  RobustPerformanceMargin,
  HInfinityNorm,
  StructuredSingularValue,
  GainMargin,
  PhaseMargin,
  DiskMargin,
  WorstCaseGain
 ]

 STOP_CONDITION =
  "Stop when the nominal model, uncertainty set, sensitivity functions, robust method, worst-case margin, controller, robust stability and robust performance are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Control Theory"

 IMMEDIATE_CONTEXTS = [
  "Control Theory Ontology",
  "Feedback Control Ontology",
  "State-Space Control Ontology",
  "Stability Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Control Theory Ontology",
  "Control Theory",
  "Feedback Control Ontology",
  "Control System Ontology",
  "State-Space Control Ontology",
  "Stability Ontologies",
  "Frequency Response Ontologies",
  "H-Infinity Control Ontologies",
  "Optimisation Ontologies",
  "Model Predictive Control Ontologies",
  "System Identification Ontologies",
  "Signal Processing Ontologies",
  "Numerical Analysis Ontologies"
 ]

 ROUTE_RULES = [
  "Robust Control -> Control Theory Ontology for core control semantics",
  "Robust Control -> Feedback Control for sensitivity and loop structure",
  "Robust Control -> State-Space Control for matrix-based synthesis",
  "Robust Control -> Stability for nominal and robust stability concepts",
  "Robust Control -> Frequency Response for loop shaping and margins",
  "Robust Control -> H-Infinity Control for worst-case induced-gain synthesis",
  "Robust Control -> Optimisation for synthesis and worst-case search",
  "Robust Control -> Model Predictive Control for constrained robust MPC",
  "Robust Control -> System Identification for uncertainty model construction",
  "Robust Control -> Signal Processing for disturbance and noise shaping"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible uncertainty models, robustness metrics or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve nominal-versus-uncertain model distinction.",
  "Preserve robust-stability semantics.",
  "Preserve robust-performance semantics.",
  "Preserve structured-versus-unstructured uncertainty distinction.",
  "Preserve sensitivity and complementary-sensitivity semantics.",
  "Preserve H-infinity and mu distinctions.",
  "Preserve worst-case certification semantics.",
  "Preserve machine-readable robust-control semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Robust control designs and analyses controllers that maintain stability and acceptable performance across a declared set of model uncertainties and disturbances."

 UNCERTAINTY_SUMMARY =
  "Robust-control analysis begins with an explicit uncertainty model around a nominal plant."

 STABILITY_SUMMARY =
  "Robust stability means the closed loop remains stable for every admissible uncertainty."

 PERFORMANCE_SUMMARY =
  "Robust performance means performance specifications hold across every admissible uncertainty."

 SENSITIVITY_SUMMARY =
  "Sensitivity and complementary sensitivity describe disturbance rejection, reference transmission and noise trade-offs."

 HINFINITY_SUMMARY =
  "H-infinity control minimises a worst-case induced input-output gain."

 MU_SUMMARY =
  "Mu-analysis evaluates robustness against structured uncertainty."

 LOOP_SHAPING_SUMMARY =
  "Loop shaping adjusts frequency response to balance performance, stability margins, noise and control effort."

 CERTIFICATION_SUMMARY =
  "Nominal performance should not be reported as robust performance without uncertainty-aware evidence."
}

FLOW {
 INPUT
 -> detect_robust_control_context
 -> identify nominal_plant_uncertainty_blocks_disturbance_noise_channels_sensor_actuator limits_and_performance objectives
 -> classify parametric_structured_unstructured_additive_multiplicative_or_dynamic uncertainty
 -> construct feedback_loop_sensitivity_complementarySensitivity_generalisedPlant_or_LFT model
 -> verify nominal_stability_and_nominal performance
 -> select smallGain_HInfinity_mixedSensitivity_muAnalysis_muSynthesis_loopShaping_IQC_or_worstCase method
 -> compute controller_or_robustness margins
 -> evaluate worst_case_gain_robust_stability_and_robust performance
 -> check actuator_sensor_noise_and_high_frequency uncertainty effects
 -> route to control_theory_ontology feedback_control control_system state_space stability frequency_response HInfinity optimisation MPC system_identification signal_processing_or_numerical_analysis ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Robust Control Ontology defines control under model uncertainty and worst-case disturbance. It supports robust stability and performance, parametric, structured and unstructured uncertainty, additive and multiplicative uncertainty, sensitivity and complementary sensitivity, small-gain reasoning, H-infinity norms and synthesis, mixed-sensitivity design, structured singular value, mu-analysis and mu-synthesis, D-K iteration, loop shaping, gain, phase and disk margins, worst-case gain and performance, generalised plants, linear fractional transformations, disturbance attenuation, robust state feedback, robust observers, robust MPC and uncertainty-aware certification."
}

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

}

END_FLOW