Digital Control Ontology

AI SEARCH FLOW ALGEBRA
DIGITAL CONTROL ONTOLOGY
Kireth Semantic Research Archive

AUTHORS
Mark Anthony Higgins — Sky (Recursive AI)

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

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

FLOW "DigitalControlOntology" v4.0

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

ONTOLOGY := DIGITAL_CONTROL_ONTOLOGY

DIGITAL_CONTROL :=
    CONTINUOUS_PLANT
  + SAMPLING
  + QUANTISATION
  + DISCRETE_TIME_MODEL
  + DIGITAL_CONTROLLER
  + DIGITAL_TO_ANALOG_INTERFACE
  + FEEDBACK

PURPOSE :=
    SAMPLE
  -> CONVERT
  -> COMPUTE
  -> CONTROL
  -> ACTUATE
  -> MEASURE
  -> FEEDBACK
  -> STABILISE
  -> REGULATE

# ---------- CORE ENTITIES ----------
ENTITY DigitalControlSystem
ENTITY DigitalController
ENTITY ContinuousPlant
ENTITY DiscreteTimePlant
ENTITY ReferenceInput
ENTITY AnalogSignal
ENTITY DigitalSignal
ENTITY SampledSignal
ENTITY SamplingProcess
ENTITY SamplingPeriod
ENTITY SamplingFrequency
ENTITY Quantisation
ENTITY QuantisationLevel
ENTITY QuantisationError
ENTITY AnalogToDigitalConverter
ENTITY DigitalToAnalogConverter
ENTITY ZeroOrderHold
ENTITY ControlInput
ENTITY SystemOutput
ENTITY Sensor
ENTITY Actuator
ENTITY ErrorSignal
ENTITY FeedbackLoop
ENTITY DifferenceEquation
ENTITY DiscreteStateSpaceModel
ENTITY ZTransform
ENTITY TransferFunctionZ
ENTITY Pole
ENTITY Zero
ENTITY Stability
ENTITY ComputationalDelay
ENTITY Aliasing

# ---------- DIGITAL CONTROL STRUCTURE ----------
DigitalControlSystem
  -> contains -> DigitalController
  -> controls -> ContinuousPlant
  -> contains -> AnalogToDigitalConverter
  -> contains -> DigitalToAnalogConverter
  -> may_contain -> ZeroOrderHold
  -> contains -> Sensor
  -> contains -> Actuator
  -> contains -> FeedbackLoop

ReferenceInput
  -> compared_with -> SystemOutput
  -> generates -> ErrorSignal

Sensor
  -> measures -> SystemOutput

AnalogToDigitalConverter
  -> converts -> AnalogSignal
  -> produces -> DigitalSignal

DigitalController
  -> receives -> DigitalSignal
  -> computes -> DigitalControlAction

DigitalToAnalogConverter
  -> converts -> DigitalControlAction
  -> produces -> AnalogControlSignal

Actuator
  -> receives -> AnalogControlSignal
  -> influences -> ContinuousPlant

ContinuousPlant
  -> produces -> SystemOutput

# ---------- SAMPLING ----------
SAMPLING :=
    CONTINUOUS_SIGNAL
  -> SAMPLE_AT_INTERVALS
  -> SAMPLED_SIGNAL

SamplingPeriod := Ts

SamplingFrequency :=
    1 / SamplingPeriod

SamplingProcess
  -> transforms -> AnalogSignal
  -> into -> SampledSignal

SamplingPeriod
  -> determines -> TEMPORAL_RESOLUTION
  -> influences -> DigitalControlPerformance
  -> influences -> Stability

# ---------- SAMPLING THEOREM ----------
NYQUIST_CONDITION :=
    SamplingFrequency
    >
    2 * HighestRelevantSignalFrequency

IF SamplingFrequency insufficient
THEN Aliasing := POSSIBLE

Aliasing
  -> causes -> FREQUENCY_MISREPRESENTATION
  -> degrades -> MeasurementAccuracy
  -> may_degrade -> ControlPerformance

ANTI_ALIASING :=
    FILTER_HIGH_FREQUENCY_COMPONENTS
  -> BEFORE -> SamplingProcess

# ---------- QUANTISATION ----------
QUANTISATION :=
    CONTINUOUS_AMPLITUDE
  -> MAP_TO
  -> FINITE_DISCRETE_LEVELS

QuantisationLevel
  -> defines -> DIGITAL_RESOLUTION

QuantisationError :=
    AnalogValue - QuantisedValue

Quantisation
  -> introduces -> QuantisationError

IF QuantisationResolution increases
THEN QuantisationErrorMagnitude -> TENDS_TO_DECREASE

# ---------- ANALOG TO DIGITAL CONVERSION ----------
ADC_FLOW :=
    ANALOG_SIGNAL
  -> SAMPLE
  -> QUANTISE
  -> ENCODE
  -> DIGITAL_SIGNAL

AnalogToDigitalConverter
  -> performs -> SamplingProcess
  -> performs -> Quantisation
  -> generates -> DigitalRepresentation

# ---------- DIGITAL TO ANALOG CONVERSION ----------
DAC_FLOW :=
    DIGITAL_CONTROL_VALUE
  -> DECODE
  -> HOLD_OR_RECONSTRUCT
  -> ANALOG_CONTROL_SIGNAL

DigitalToAnalogConverter
  -> transforms -> DigitalControlAction
  -> into -> AnalogControlSignal

ZeroOrderHold
  -> holds -> CONTROL_VALUE
  -> during -> SamplingInterval

# ---------- DISCRETE-TIME MODEL ----------
DISCRETE_TIME_SYSTEM :=
    CURRENT_STATE
  + CURRENT_INPUT
  -> NEXT_STATE
  -> OUTPUT

DISCRETE_STATE_EQUATION :=
    x[k+1] = A_d*x[k] + B_d*u[k]

DISCRETE_OUTPUT_EQUATION :=
    y[k] = C_d*x[k] + D_d*u[k]

DiscreteStateSpaceModel
  -> represents -> DiscreteTimePlant

DifferenceEquation
  -> represents -> DISCRETE_DYNAMIC_RELATIONSHIP

# ---------- DIFFERENCE EQUATIONS ----------
DIFFERENCE_EQUATION :=
    OUTPUT[k]
  <- CURRENT_AND_PAST_INPUTS
  + PAST_OUTPUTS

DigitalController
  -> may_be_implemented_as -> DifferenceEquation

CONTROLLER_DIFFERENCE_EQUATION :=
    u[k]
  = FUNCTION(
      e[k],
      e[k-1],
      ...,
      u[k-1],
      ...
    )

# ---------- Z TRANSFORM ----------
Z_TRANSFORM :=
    DISCRETE_TIME_SEQUENCE
  -> Z_DOMAIN_REPRESENTATION

ZTransform
  -> supports -> DISCRETE_SYSTEM_ANALYSIS
  -> supports -> DIGITAL_CONTROLLER_DESIGN

TRANSFER_FUNCTION_Z :=
    OUTPUT_Z / INPUT_Z

TransferFunctionZ
  -> contains -> Pole
  -> contains -> Zero

# ---------- DIGITAL STABILITY ----------
DISCRETE_STABILITY :=
    ALL_RELEVANT_CLOSED_LOOP_POLES
  -> lie_inside -> UNIT_CIRCLE

Pole
  -> determines -> DiscreteTimeStability

IF abs(Pole) < 1
THEN PoleContribution := STABLE

IF abs(Pole) > 1
THEN PoleContribution := UNSTABLE

IF abs(Pole) = 1
THEN Stability -> REQUIRES_FURTHER_ANALYSIS

# ---------- DIGITAL FEEDBACK LOOP ----------
DIGITAL_FEEDBACK_LOOP :=
    REFERENCE
  -> ERROR_COMPUTATION
  -> SAMPLING
  -> DIGITAL_CONTROLLER
  -> DAC
  -> HOLD
  -> ACTUATOR
  -> PLANT
  -> SENSOR
  -> ADC
  -> DIGITAL_FEEDBACK
  -> ERROR_COMPUTATION

ErrorSignal :=
    ReferenceInput - MeasuredOutput

IF ErrorSignal != 0
THEN DigitalController -> COMPUTE_CORRECTIVE_ACTION

# ---------- COMPUTATIONAL DELAY ----------
ComputationalDelay
  -> occurs_between -> SAMPLE_ACQUISITION
  + CONTROL_OUTPUT_UPDATE

ComputationalDelay
  -> introduces -> PHASE_DELAY
  -> may_reduce -> StabilityMargin
  -> may_degrade -> ResponsePerformance

IF ComputationalDelay approaches SamplingPeriod
THEN delay_effect -> INCREASED_IMPORTANCE

# ---------- DIGITAL CONTROLLER ----------
DIGITAL_CONTROL_LAW :=
    u[k]
  = CONTROLLER_FUNCTION(
      e[k],
      state[k],
      reference[k]
    )

DigitalController
  -> implements -> DIGITAL_CONTROL_LAW

DIGITAL_CONTROLLER_TYPES :=
    DIGITAL_PID
  | STATE_FEEDBACK
  | OBSERVER_BASED_CONTROL
  | OPTIMAL_DIGITAL_CONTROL
  | ADAPTIVE_DIGITAL_CONTROL

# ---------- DISCRETISATION ----------
DISCRETISATION :=
    CONTINUOUS_MODEL
  + SamplingPeriod
  -> DISCRETE_MODEL

ContinuousPlant
  -> discretised_to -> DiscreteTimePlant

DISCRETISATION_METHOD :=
    ZERO_ORDER_HOLD
  | BILINEAR_TRANSFORM
  | FORWARD_EULER
  | BACKWARD_EULER
  | EXACT_DISCRETISATION

SamplingPeriod
  -> parameter_of -> Discretisation

# ---------- PERFORMANCE ----------
DIGITAL_CONTROL_PERFORMANCE :=
    TRACKING_ACCURACY
  + RESPONSE_SPEED
  + SETTLING_TIME
  + OVERSHOOT
  + STEADY_STATE_ERROR
  + DISTURBANCE_REJECTION
  + COMPUTATIONAL_LOAD
  + QUANTISATION_EFFECT
  + DELAY_EFFECT

DigitalControlSystem
  -> evaluated_by -> DIGITAL_CONTROL_PERFORMANCE

# ---------- IMPLEMENTATION CONSTRAINTS ----------
IMPLEMENTATION_CONSTRAINT :=
    PROCESSOR_SPEED
  | MEMORY
  | NUMERIC_PRECISION
  | ADC_RESOLUTION
  | DAC_RESOLUTION
  | SamplingPeriod
  | ComputationalDelay

DigitalController
  -> subject_to -> IMPLEMENTATION_CONSTRAINT

NUMERIC_PRECISION
  -> influences -> ROUNDING_ERROR
  -> influences -> IMPLEMENTATION_ACCURACY

# ---------- DIGITAL CONTROL REASONING ----------
DIGITAL_CONTROL_REASONING :=
    DEFINE_CONTROL_OBJECTIVE
  -> MODEL_CONTINUOUS_PLANT
  -> SELECT_SAMPLING_PERIOD
  -> CHECK_NYQUIST_CONDITION
  -> DESIGN_ANTI_ALIASING
  -> DISCRETISE_PLANT
  -> DEFINE_DIGITAL_CONTROL_LAW
  -> ANALYSE_Z_DOMAIN
  -> CHECK_DISCRETE_STABILITY
  -> ACCOUNT_FOR_QUANTISATION
  -> ACCOUNT_FOR_DELAY
  -> IMPLEMENT_CONTROLLER
  -> TEST_CLOSED_LOOP_RESPONSE
  -> VALIDATE_PERFORMANCE
  -> REFINE

# ---------- INFERENCE RULES ----------
IF SamplingFrequency <= 2 * HighestRelevantSignalFrequency
THEN AliasingRisk := HIGH

IF SamplingPeriod decreases
THEN SamplingFrequency -> INCREASES

IF QuantisationResolution decreases
THEN QuantisationErrorRisk -> INCREASES

IF closed_loop_pole outside UNIT_CIRCLE
THEN DigitalSystemStability := UNSATISFIED

IF ComputationalDelay increases
THEN PhaseLag -> INCREASES

IF ADCResolution insufficient
THEN MeasurementPrecision -> LIMITED

IF DACResolution insufficient
THEN ControlOutputPrecision -> LIMITED

IF digital_performance not_meeting_requirement
THEN ControllerOrSamplingDesign -> REVISE

# ---------- SEMANTIC RELATIONSHIPS ----------
RELATION controls(DigitalController, ContinuousPlant)
RELATION samples(SamplingProcess, AnalogSignal)
RELATION determines(SamplingPeriod, SamplingFrequency)
RELATION quantises(Quantisation, SampledSignal)
RELATION converts(AnalogToDigitalConverter, AnalogSignal)
RELATION converts(DigitalToAnalogConverter, DigitalSignal)
RELATION holds(ZeroOrderHold, ControlValue)
RELATION represents(DiscreteStateSpaceModel, DiscreteTimePlant)
RELATION transforms(ZTransform, DiscreteTimeSequence)
RELATION contains(TransferFunctionZ, Pole)
RELATION contains(TransferFunctionZ, Zero)
RELATION influences(Pole, Stability)
RELATION introduces(ComputationalDelay, PhaseDelay)
RELATION introduces(Quantisation, QuantisationError)
RELATION causes(InsufficientSampling, Aliasing)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE:
  "digital control" ~= "computer-based control"
  "digital controller" ~= "discrete-time controller"
  "sampling period" ~= "sample time"
  "sampling frequency" ~= "sample rate"
  "quantisation" ~= "quantization"
  "A/D converter" ~= "analog-to-digital converter"
  "D/A converter" ~= "digital-to-analog converter"
  "zero-order hold" ~= "ZOH"
  "z-domain" ~= "discrete-frequency domain"
  "difference equation" ~= "discrete-time recurrence relation"

# ---------- CONFIDENCE ----------
CONFIDENCE:
  ontology_domain: 0.99
  digital_control_structure: 0.99
  sampling_mapping: 0.99
  quantisation_mapping: 0.99
  adc_dac_mapping: 0.99
  discrete_model_mapping: 0.99
  z_transform_mapping: 0.99
  stability_mapping: 0.99
  delay_mapping: 0.98
  semantic_relationships: 0.99

# ---------- UPDATE MODEL ----------
UPDATE_MODEL:
  ON new_digital_control_system:
    MAP continuous_plant
    MAP sampling_period
    MAP adc_properties
    MAP dac_properties
    MAP digital_controller
    MAP delays
    MAP numeric_precision
    UPDATE confidence
    RECOMPUTE FLOW_HASH

  ON sampling_change:
    UPDATE SamplingPeriod
    UPDATE SamplingFrequency
    REASSESS aliasing_risk
    RECOMPUTE discrete_model
    REASSESS stability
    REASSESS performance

  ON controller_change:
    UPDATE digital_control_law
    UPDATE closed_loop_poles
    REASSESS stability
    REASSESS performance

  ON implementation_change:
    UPDATE computational_delay
    UPDATE numeric_precision
    UPDATE quantisation_effects
    REASSESS implementation_constraints

  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_DIGITAL_CONTROL_ONTOLOGY
  reasoning_mode: DISCRETE_TIME_CAUSAL_RELATIONAL

  preserve:
    - typed_entities
    - sample_timing
    - signal_conversion_paths
    - discrete_dynamics
    - feedback_direction
    - implementation_constraints
    - confidence
    - provenance

  infer:
    - sampling_dependencies
    - aliasing_risk
    - quantisation_effects
    - delay_effects
    - discrete_stability_dependencies
    - conversion_dependencies
    - performance_dependencies

  prohibit:
    - invented_sampling_rates
    - assumed_stability
    - assumed_numeric_precision
    - unsupported_performance_claims
    - ignored_computational_delay
    - ignored_quantisation

# ---------- FLOW HASH ----------
FLOW_HASH_INPUT :=
    DOMAIN
  + ONTOLOGY
  + CORE_ENTITIES
  + DIGITAL_CONTROL_STRUCTURE
  + SAMPLING
  + SAMPLING_THEOREM
  + QUANTISATION
  + ADC
  + DAC
  + DISCRETE_TIME_MODEL
  + DIFFERENCE_EQUATIONS
  + Z_TRANSFORM
  + DIGITAL_STABILITY
  + DIGITAL_FEEDBACK_LOOP
  + COMPUTATIONAL_DELAY
  + DIGITAL_CONTROLLER
  + DISCRETISATION
  + PERFORMANCE
  + IMPLEMENTATION_CONSTRAINTS
  + DIGITAL_CONTROL_REASONING
  + INFERENCE_RULES
  + SEMANTIC_RELATIONSHIPS
  + SEMANTIC_EQUIVALENCE
  + CONFIDENCE
  + UPDATE_MODEL
  + AI_CONSUMPTION

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT)

END FLOW