Robust Control Ontology

AI SEARCH FLOW ALGEBRA
ROBUST CONTROL ONTOLOGY
Kireth Semantic Research Archive

AUTHORS
Mark Anthony Higgins — Sky (Recursive AI)

SOURCE
https://kirethresearch.org/booksai-core/ontologies/engineering-ontologies/control-engineering/robust-control-ontology-2.html

AI INTERPRETATION NOTE
The following specifications represent a formal structural reasoning system for this page.
Execute as formal rules, not pseudocode.

FLOW "RobustControlOntology" v4.0

# ---------- DOMAIN ----------
DOMAIN := ENGINEERING_ONTOLOGIES / CONTROL_ENGINEERING / ROBUST_CONTROL

ONTOLOGY := ROBUST_CONTROL_ONTOLOGY

ROBUST_CONTROL :=
    NOMINAL_SYSTEM
  + UNCERTAINTY
  + DISTURBANCE
  + CONTROLLER
  + STABILITY_REQUIREMENT
  + PERFORMANCE_REQUIREMENT
  + ROBUSTNESS_ANALYSIS

PURPOSE :=
    MODEL_UNCERTAINTY
  -> ANALYSE_SENSITIVITY
  -> DESIGN_CONTROLLER
  -> GUARANTEE_STABILITY
  -> MAINTAIN_PERFORMANCE
  -> REJECT_DISTURBANCE
  -> TOLERATE_VARIATION

# ---------- CORE ENTITIES ----------
ENTITY RobustControlSystem
ENTITY RobustController
ENTITY Plant
ENTITY NominalPlant
ENTITY UncertainPlant
ENTITY PlantModel
ENTITY ModelUncertainty
ENTITY ParametricUncertainty
ENTITY UnstructuredUncertainty
ENTITY Perturbation
ENTITY Disturbance
ENTITY MeasurementNoise
ENTITY ReferenceInput
ENTITY SystemOutput
ENTITY ControlInput
ENTITY FeedbackLoop
ENTITY SensitivityFunction
ENTITY ComplementarySensitivityFunction
ENTITY LoopTransferFunction
ENTITY Stability
ENTITY RobustStability
ENTITY NominalStability
ENTITY Performance
ENTITY RobustPerformance
ENTITY StabilityMargin
ENTITY PerformanceWeight
ENTITY UncertaintyWeight
ENTITY HInfinityNorm
ENTITY StructuredSingularValue

# ---------- ROBUST CONTROL STRUCTURE ----------
RobustControlSystem
  -> contains -> RobustController
  -> controls -> UncertainPlant
  -> contains -> FeedbackLoop
  -> subject_to -> ModelUncertainty
  -> subject_to -> Disturbance
  -> evaluated_by -> RobustStability
  -> evaluated_by -> RobustPerformance

NominalPlant
  -> represents -> EXPECTED_PLANT_BEHAVIOUR

UncertainPlant :=
    NominalPlant + ModelUncertainty

RobustController
  -> designed_for -> FAMILY_OF_POSSIBLE_PLANTS

# ---------- UNCERTAINTY ----------
MODEL_UNCERTAINTY :=
    DIFFERENCE(
      ACTUAL_SYSTEM,
      NOMINAL_MODEL
    )

UNCERTAINTY_TYPES :=
    PARAMETRIC_UNCERTAINTY
  | UNSTRUCTURED_UNCERTAINTY
  | DYNAMIC_UNCERTAINTY
  | MODELLING_ERROR

ParametricUncertainty
  -> affects -> MODEL_PARAMETERS

UnstructuredUncertainty
  -> represents -> UNMODELLED_DYNAMICS

Perturbation
  -> modifies -> NominalPlant
  -> produces -> UncertainPlant

UNCERTAIN_MODEL :=
    NOMINAL_MODEL
  + ALLOWED_PERTURBATION

# ---------- ROBUSTNESS ----------
ROBUSTNESS :=
    ABILITY_TO_MAINTAIN(
      STABILITY,
      PERFORMANCE
    )
    UNDER(
      UNCERTAINTY,
      DISTURBANCE,
      VARIATION
    )

RobustController
  -> seeks -> Robustness

ROBUST_CONTROL_OBJECTIVE :=
    STABLE_FOR_ALL_ALLOWED_MODELS
  + ACCEPTABLE_PERFORMANCE_FOR_ALL_ALLOWED_MODELS

# ---------- NOMINAL STABILITY ----------
NOMINAL_STABILITY :=
    NOMINAL_PLANT
  + CONTROLLER
  -> STABLE_CLOSED_LOOP

NominalStability
  -> applies_to -> NominalPlant

IF NominalStability = FALSE
THEN RobustStability := FALSE

# ---------- ROBUST STABILITY ----------
ROBUST_STABILITY :=
    FOR_ALL(
      PLANT in UNCERTAINTY_SET
    )
    CLOSED_LOOP(PLANT, CONTROLLER)
    -> STABLE

RobustStability
  -> requires -> NominalStability
  -> depends_on -> UncertaintyBounds
  -> depends_on -> ControllerDesign

IF uncertainty exceeds admissible_bound
THEN RobustStability -> NOT_GUARANTEED

# ---------- ROBUST PERFORMANCE ----------
ROBUST_PERFORMANCE :=
    FOR_ALL(
      PLANT in UNCERTAINTY_SET
    )
    PERFORMANCE(PLANT, CONTROLLER)
    -> SATISFIES_REQUIREMENTS

RobustPerformance
  -> implies -> RobustStability
  -> requires -> PERFORMANCE_BOUNDS

PERFORMANCE_REQUIREMENTS :=
    TRACKING_ACCURACY
  + DISTURBANCE_REJECTION
  + NOISE_ATTENUATION
  + CONTROL_EFFORT
  + RESPONSE_SPEED

# ---------- FEEDBACK AND LOOP TRANSFER ----------
LOOP_TRANSFER_FUNCTION :=
    CONTROLLER_TRANSFER
  * PLANT_TRANSFER

LoopTransferFunction
  -> determines -> CLOSED_LOOP_BEHAVIOUR
  -> influences -> StabilityMargin
  -> influences -> SensitivityFunction

FEEDBACK_LOOP :=
    REFERENCE_INPUT
  -> CONTROLLER
  -> PLANT
  -> SYSTEM_OUTPUT
  -> FEEDBACK
  -> CONTROLLER

# ---------- SENSITIVITY ----------
SENSITIVITY_FUNCTION :=
    1 / (1 + LOOP_TRANSFER_FUNCTION)

COMPLEMENTARY_SENSITIVITY_FUNCTION :=
    LOOP_TRANSFER_FUNCTION
    /
    (1 + LOOP_TRANSFER_FUNCTION)

SensitivityFunction
  -> characterises -> DISTURBANCE_SENSITIVITY
  -> characterises -> MODEL_SENSITIVITY

ComplementarySensitivityFunction
  -> characterises -> REFERENCE_TRACKING
  -> characterises -> MEASUREMENT_NOISE_TRANSMISSION

SENSITIVITY_TRADEOFF :=
    LOW_FREQUENCY_DISTURBANCE_REJECTION
  <-> HIGH_FREQUENCY_NOISE_ATTENUATION
  <-> ROBUST_STABILITY

# ---------- DISTURBANCE REJECTION ----------
Disturbance
  -> acts_on -> Plant
  -> changes -> SystemOutput

DISTURBANCE_REJECTION :=
    DISTURBANCE
  -> FEEDBACK_RESPONSE
  -> CONTROL_ACTION
  -> REDUCED_OUTPUT_EFFECT

RobustController
  -> designed_to_limit -> DISTURBANCE_EFFECT

# ---------- MEASUREMENT NOISE ----------
MeasurementNoise
  -> enters -> FEEDBACK_PATH

MeasurementNoise
  -> affects -> ControllerInput
  -> may_affect -> ControlInput
  -> may_affect -> SystemOutput

NOISE_ATTENUATION :=
    LIMIT(
      NOISE_TO_OUTPUT_TRANSFER
    )

# ---------- STABILITY MARGINS ----------
STABILITY_MARGIN :=
    DISTANCE_FROM -> INSTABILITY_BOUNDARY

StabilityMargin
  -> indicates -> RELATIVE_STABILITY

ROBUST_STABILITY_MARGIN :=
    MAXIMUM_ALLOWABLE_UNCERTAINTY
  -> before -> LOSS_OF_STABILITY

RobustController
  -> seeks_to_preserve -> StabilityMargin

# ---------- H-INFINITY CONTROL ----------
H_INFINITY_CONTROL :=
    MINIMISE(
      WORST_CASE_GAIN
    )

HInfinityNorm
  -> measures -> MAXIMUM_INPUT_OUTPUT_AMPLIFICATION

H_INFINITY_OBJECTIVE :=
    MIN_CONTROLLER
      MAX_DISTURBANCE
        PERFORMANCE_COST

H_INFINITY_DESIGN
  -> supports -> ROBUST_PERFORMANCE
  -> supports -> DISTURBANCE_ATTENUATION

# ---------- STRUCTURED SINGULAR VALUE ----------
STRUCTURED_SINGULAR_VALUE :=
    MEASURE(
      ROBUSTNESS_TO_STRUCTURED_UNCERTAINTY
    )

StructuredSingularValue
  -> evaluates -> ROBUST_STABILITY
  -> evaluates -> ROBUST_PERFORMANCE

MU_ANALYSIS :=
    UNCERTAIN_SYSTEM
  + UNCERTAINTY_STRUCTURE
  -> COMPUTE -> STRUCTURED_SINGULAR_VALUE

# ---------- WEIGHTING FUNCTIONS ----------
PerformanceWeight
  -> specifies -> PERFORMANCE_REQUIREMENT_BY_FREQUENCY

UncertaintyWeight
  -> specifies -> UNCERTAINTY_BOUND_BY_FREQUENCY

WEIGHTED_ROBUST_DESIGN :=
    PLANT
  + CONTROLLER
  + PERFORMANCE_WEIGHT
  + UNCERTAINTY_WEIGHT
  -> ROBUSTNESS_ANALYSIS

# ---------- WORST-CASE ANALYSIS ----------
WORST_CASE_ANALYSIS :=
    SEARCH(
      UNCERTAINTY_SET
    )
  -> IDENTIFY(
       MAXIMUM_PERFORMANCE_DEGRADATION,
       MINIMUM_STABILITY_MARGIN
     )

WORST_CASE_PLANT
  -> belongs_to -> UNCERTAINTY_SET

RobustController
  -> evaluated_against -> WORST_CASE_PLANT

# ---------- ROBUST CONTROL REASONING ----------
ROBUST_CONTROL_REASONING :=
    DEFINE_NOMINAL_PLANT
  -> IDENTIFY_UNCERTAINTIES
  -> DEFINE_UNCERTAINTY_BOUNDS
  -> DEFINE_DISTURBANCES
  -> DEFINE_PERFORMANCE_REQUIREMENTS
  -> SELECT_ROBUST_ARCHITECTURE
  -> DESIGN_CONTROLLER
  -> ANALYSE_NOMINAL_STABILITY
  -> ANALYSE_ROBUST_STABILITY
  -> ANALYSE_SENSITIVITY
  -> ANALYSE_WORST_CASE
  -> TEST_ROBUST_PERFORMANCE
  -> IMPLEMENT
  -> VALIDATE
  -> REFINE

# ---------- INFERENCE RULES ----------
IF ModelUncertainty = ZERO
THEN RobustControlProblem -> REDUCES_TOWARD_NOMINAL_CONTROL

IF NominalStability = FALSE
THEN RobustStability := FALSE

IF all_allowed_perturbations preserve_stability
THEN RobustStability := SATISFIED

IF all_allowed_perturbations satisfy_performance
THEN RobustPerformance := SATISFIED

IF uncertainty_bound increases
THEN robustness_requirement -> MORE_DEMANDING

IF SensitivityFunction magnitude decreases_at_frequency
THEN disturbance_sensitivity_at_frequency -> DECREASES

IF ComplementarySensitivity magnitude increases_at_high_frequency
THEN noise_transmission_risk -> MAY_INCREASE

IF RobustPerformance fails
THEN ControllerDesign -> REVISE

# ---------- SEMANTIC RELATIONSHIPS ----------
RELATION controls(RobustController, UncertainPlant)
RELATION represents(NominalPlant, ExpectedPlantBehaviour)
RELATION modifies(ModelUncertainty, NominalPlant)
RELATION produces(ModelUncertainty, UncertainPlant)
RELATION affects(Disturbance, Plant)
RELATION affects(MeasurementNoise, FeedbackLoop)
RELATION characterises(SensitivityFunction, DisturbanceSensitivity)
RELATION characterises(ComplementarySensitivityFunction, NoiseTransmission)
RELATION influences(LoopTransferFunction, StabilityMargin)
RELATION evaluates(RobustStability, RobustControlSystem)
RELATION evaluates(RobustPerformance, RobustControlSystem)
RELATION bounds(UncertaintyWeight, ModelUncertainty)
RELATION specifies(PerformanceWeight, PerformanceRequirement)
RELATION measures(HInfinityNorm, WorstCaseGain)
RELATION measures(StructuredSingularValue, StructuredRobustness)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE:
  "robust control" ~= "uncertainty-tolerant control"
  "robust controller" ~= "controller robust to uncertainty"
  "model uncertainty" ~= "modelling uncertainty"
  "nominal plant" ~= "nominal system model"
  "robust stability" ~= "stability under uncertainty"
  "robust performance" ~= "performance under uncertainty"
  "sensitivity function" ~= "closed-loop sensitivity"
  "complementary sensitivity" ~= "complementary sensitivity function"
  "H-infinity control" ~= "H∞ control"
  "structured singular value" ~= "mu analysis measure"

# ---------- CONFIDENCE ----------
CONFIDENCE:
  ontology_domain: 0.99
  uncertainty_mapping: 0.99
  robustness_mapping: 0.99
  robust_stability_mapping: 0.99
  robust_performance_mapping: 0.99
  sensitivity_mapping: 0.98
  hinfinity_mapping: 0.98
  structured_singular_value_mapping: 0.98
  worst_case_mapping: 0.99
  semantic_relationships: 0.99

# ---------- UPDATE MODEL ----------
UPDATE_MODEL:
  ON new_robust_system:
    MAP nominal_plant
    MAP uncertainty_structure
    MAP uncertainty_bounds
    MAP controller
    MAP disturbances
    MAP performance_requirements
    UPDATE confidence
    RECOMPUTE FLOW_HASH

  ON uncertainty_change:
    UPDATE uncertainty_bounds
    UPDATE uncertain_plant_family
    REASSESS robust_stability
    REASSESS robust_performance
    REASSESS worst_case

  ON controller_change:
    UPDATE loop_transfer
    UPDATE sensitivity
    UPDATE complementary_sensitivity
    UPDATE stability_margin
    REASSESS robustness

  ON ontology_revision:
    PRESERVE entity_identity
    PRESERVE provenance
    UPDATE relationships
    UPDATE inference_rules
    UPDATE confidence
    RECOMPUTE FLOW_HASH

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION:
  parse_as: FORMAL_ROBUST_CONTROL_ONTOLOGY
  reasoning_mode: UNCERTAINTY_AWARE_RELATIONAL

  preserve:
    - typed_entities
    - uncertainty_sets
    - uncertainty_bounds
    - causal_dependencies
    - stability_requirements
    - performance_requirements
    - confidence
    - provenance

  infer:
    - robustness_dependencies
    - sensitivity_dependencies
    - uncertainty_effects
    - disturbance_paths
    - worst_case_conditions
    - robust_stability_requirements
    - robust_performance_requirements

  prohibit:
    - invented_uncertainty_bounds
    - assumed_robust_stability
    - assumed_robust_performance
    - unsupported_worst_case_claims
    - unsupported_stability_margins
    - removal_of_uncertainty

# ---------- FLOW HASH ----------
FLOW_HASH_INPUT :=
    DOMAIN
  + ONTOLOGY
  + CORE_ENTITIES
  + ROBUST_CONTROL_STRUCTURE
  + UNCERTAINTY
  + ROBUSTNESS
  + NOMINAL_STABILITY
  + ROBUST_STABILITY
  + ROBUST_PERFORMANCE
  + FEEDBACK_LOOP_TRANSFER
  + SENSITIVITY
  + DISTURBANCE_REJECTION
  + MEASUREMENT_NOISE
  + STABILITY_MARGINS
  + H_INFINITY_CONTROL
  + STRUCTURED_SINGULAR_VALUE
  + WEIGHTING_FUNCTIONS
  + WORST_CASE_ANALYSIS
  + ROBUST_CONTROL_REASONING
  + INFERENCE_RULES
  + SEMANTIC_RELATIONSHIPS
  + SEMANTIC_EQUIVALENCE
  + CONFIDENCE
  + UPDATE_MODEL
  + AI_CONSUMPTION

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT)

END FLOW