AI INTERPRETATION NOTE
This Flow Algebra defines the Control Theory Ontology within the Kireth Research Semantic Library. It formalises control theory as a machine-readable ontology of dynamical systems, plants, inputs, outputs, feedback, state variables, transfer functions, state-space models, stability, controllability, observability, controller design, estimation, optimal control, robust control, adaptive control and model predictive control. It sits beneath the Control Theory parent layer and provides a structured reasoning framework for representing, classifying, analysing, controlling and validating dynamical systems.
FLOW_ALGEBRA_RECORD {
METADATA {
FLOW_NAME = "Control Theory Ontology"
FLOW_TYPE = "Mathematical Ontology"
DOMAIN = "Mathematics"
SUBDOMAIN = "Applied Mathematics"
SPECIALISATION = "Control Theory / Core Control Theory Ontology"
CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/control-theory/control-theory-ontology.html"
LANGUAGE = "en-GB"
VERSION = "1.0"
STATUS = "active"
}
IDENTITY {
PRIMARY_ENTITY = ControlTheoryOntology
ENTITY_CLASS = FormalDynamicalControlOntology
PRIMARY_CONTEXT = ControlTheory
DEFINITION = "The Control Theory Ontology is a formal semantic framework for representing dynamical systems, control inputs, outputs, feedback, state-space and transfer-function models, stability, controllability, observability, estimation, controller structures and closed-loop performance."
SEMANTIC_ROLE = "Provides machine-readable entities and reasoning relations for analysing system dynamics, classifying control problems, designing controllers and estimators, and validating closed-loop behaviour."
}
SCOPE {
INCLUDE = [
ControlTheoryOntology,
ControlSystem,
DynamicalSystem,
Plant,
Controller,
ControlInput,
ManipulatedVariable,
Disturbance,
ReferenceSignal,
MeasuredOutput,
ControlledOutput,
StateVariable,
StateVector,
Feedback,
NegativeFeedback,
PositiveFeedback,
Feedforward,
OpenLoopSystem,
ClosedLoopSystem,
TransferFunction,
StateSpaceModel,
StateEquation,
OutputEquation,
LinearSystem,
NonlinearSystem,
LinearTimeInvariantSystem,
TimeVaryingSystem,
ContinuousTimeSystem,
DiscreteTimeSystem,
Stability,
LyapunovStability,
AsymptoticStability,
ExponentialStability,
BIBOStability,
MarginalStability,
Controllability,
Reachability,
Stabilizability,
Observability,
Detectability,
Pole,
Zero,
PolePlacement,
FrequencyResponse,
GainMargin,
PhaseMargin,
BodePlot,
NyquistCriterion,
RootLocus,
PIDControl,
StateFeedback,
Observer,
LuenbergerObserver,
KalmanFilter,
OptimalControl,
LinearQuadraticRegulator,
RobustControl,
HInfinityControl,
AdaptiveControl,
NonlinearControl,
StochasticControl,
ModelPredictiveControl,
StateEstimation,
SystemIdentification,
Tracking,
Regulation,
DisturbanceRejection,
ControlEffort,
ActuatorSaturation
]
EXCLUDE_AS_PRIMARY = [
PureDynamicalSystemsAnalysisWithoutControlObjective,
SignalProcessingWithoutControlLoopSemantics,
GenericOptimisationWithoutSystemDynamics,
MechanicalOrElectricalDesignWithoutControlLawContext
]
DISAMBIGUATION_RULE =
"When a dynamical system is modelled, analysed, estimated or influenced using control inputs and feedback to achieve desired behaviour, interpret the context through the Control Theory Ontology."
}
CORE_CONCEPTS {
ControlTheoryOntology {
definition = "The ontology organising the formal entities and relations of control theory."
}
ControlSystem {
definition = "A dynamical system together with control inputs, measurements and a controller."
}
Plant {
definition = "The process or system whose behaviour is to be controlled."
}
Controller {
definition = "A rule or mechanism computing control actions from references, measurements or estimated states."
}
StateVariable {
definition = "A variable that contributes to a minimal internal description of the system dynamics."
}
Feedback {
definition = "A structure in which measured or estimated system behaviour affects subsequent control actions."
}
TransferFunction {
definition = "An input-output representation of a linear time-invariant system in the Laplace or z domain under declared assumptions."
}
StateSpaceModel {
definition = "A representation of system dynamics using a state vector and first-order evolution equations."
}
Stability {
definition = "A property describing boundedness or convergence of system responses under perturbations."
}
Controllability {
definition = "The ability to steer the system state through admissible control inputs."
}
Observability {
definition = "The ability to reconstruct the internal state from output measurements over time."
}
ClosedLoopPerformance {
definition = "The achieved tracking, regulation, robustness, speed, damping, effort and disturbance-rejection behaviour of the controlled system."
}
}
RELATIONS {
ControlTheoryOntology --is_child_of--> ControlTheory
ControlSystem --contains--> Plant
ControlSystem --contains--> Controller
Controller --produces--> ControlInput
ControlInput --acts_on--> Plant
Disturbance --acts_on--> Plant
Plant --produces--> MeasuredOutput
Feedback --uses--> MeasuredOutput
Feedback --influences--> Controller
StateSpaceModel --represents--> DynamicalSystem
TransferFunction --represents--> LinearTimeInvariantSystem
Pole --influences--> Stability
Zero --influences--> InputOutputBehaviour
Controllability --supports--> StateFeedback
Observability --supports--> StateEstimation
Stabilizability --supports--> StableStateFeedback
Detectability --supports--> StableObserverDesign
Observer --estimates--> StateVector
KalmanFilter --is_type_of--> StateEstimator
PIDControl --is_type_of--> FeedbackController
LinearQuadraticRegulator --is_type_of--> OptimalControl
ModelPredictiveControl --uses--> RecedingHorizonOptimisation
RobustControl --addresses--> ModelUncertainty
AdaptiveControl --responds_to--> ParameterVariation
}
OPERATORS {
CONTINUOUS_STATE_MODEL(A,B,C,D,x,u) :=
x_dot=A*x+B*u
AND y=C*x+D*u
DISCRETE_STATE_MODEL(A,B,C,D,x_k,u_k) :=
x_{k+1}=A*x_k+B*u_k
AND y_k=C*x_k+D*u_k
STATE_FEEDBACK(K,x) :=
u=-K*x
CLOSED_LOOP_MATRIX(A,B,K) :=
A-B*K
OBSERVER_DYNAMICS(A,B,C,L,xhat,u,y) :=
xhat_dot=A*xhat+B*u+L*(y-C*xhat)
CONTROLLABILITY_MATRIX(A,B) :=
[B, A*B, A^2*B, ..., A^(n-1)*B]
OBSERVABILITY_MATRIX(A,C) :=
[C; C*A; C*A^2; ...; C*A^(n-1)]
CONTROLLABLE(A,B) :=
rank(CONTROLLABILITY_MATRIX(A,B))=n
OBSERVABLE(A,C) :=
rank(OBSERVABILITY_MATRIX(A,C))=n
TRANSFER_FUNCTION(A,B,C,D,s) :=
C*(sI-A)^(-1)*B + D
PID_CONTROL(e,Kp,Ki,Kd) :=
Kp*e + Ki*integral(e dt) + Kd*de/dt
TRACKING_ERROR(r,y) :=
e=r-y
LQR_COST(x,u,Q,R) :=
integral (x^T*Q*x + u^T*R*u) dt
LYAPUNOV_EQUATION(A,P,Q) :=
A^T*P+P*A=-Q
}
DEFINITIONS {
D1 = "Control systems combine a plant, one or more control inputs and a control law."
D2 = "Feedback distinguishes closed-loop from open-loop control."
D3 = "State-space models expose internal dynamics."
D4 = "Transfer functions expose linear input-output dynamics."
D5 = "Stability is central to controller design."
D6 = "Controllability and stabilizability govern what state-feedback can achieve."
D7 = "Observability and detectability govern what state estimation can achieve."
D8 = "Controller performance includes tracking, regulation, disturbance rejection and control effort."
D9 = "Robust control addresses uncertainty explicitly."
D10 = "Model predictive control solves constrained optimisation repeatedly online."
}
AXIOMS {
A1_CONTROL_STRUCTURE:
ControlTheoryOntology represents a dynamical system together with one or more control or estimation mechanisms
A2_STATE_EVOLUTION:
System behaviour evolves according to continuous, discrete or hybrid dynamics
A3_FEEDBACK:
ClosedLoopSystem contains a path from measured or estimated behaviour back to control action
A4_STABILITY:
Controller design should preserve or establish appropriate closed-loop stability
A5_CONTROLLABILITY:
Arbitrary state placement by state feedback requires appropriate controllability assumptions
A6_OBSERVABILITY:
Arbitrary state reconstruction requires appropriate observability assumptions
A7_MODEL_DEPENDENCE:
Analysis and design require a model or an explicitly declared data-driven surrogate
A8_PERFORMANCE:
Control solutions are evaluated against declared behavioural objectives and constraints
}
REASONING {
R1_ENTITY_EXTRACTION {
INPUT = control_problem_description
IDENTIFY = plant
IDENTIFY = state_variables
IDENTIFY = control_inputs
IDENTIFY = outputs
IDENTIFY = measurements
IDENTIFY = references
IDENTIFY = disturbances
RETURN = structured_control_entities
}
R2_SYSTEM_CLASSIFICATION {
INPUT = structured_control_entities
CLASSIFY = continuous_or_discrete
CLASSIFY = linear_or_nonlinear
CLASSIFY = time_invariant_or_time_varying
CLASSIFY = deterministic_or_stochastic
RETURN = control_system_class
}
R3_MODEL_SELECTION {
INPUT = system_class_and_available information
IF LTI_input_output model available
SELECT = TransferFunction
IF internal_state model available
SELECT = StateSpaceModel
IF nonlinear dynamics dominate
SELECT = NonlinearStateModel
RETURN = control_model
}
R4_STABILITY_REASONING {
INPUT = open_loop_or_closed_loop model
IF continuous_LTI
ANALYSE = eigenvalues_or_poles
IF discrete_LTI
ANALYSE = eigenvalues_relative_to_unit circle
IF nonlinear
CONSIDER = Lyapunov function
RETURN = stability_profile
}
R5_CONTROLLABILITY_REASONING {
INPUT = A_B_or_general model
IF LTI
COMPUTE = controllability_matrix rank
ELSE
CONSIDER = nonlinear reachability tools
RETURN = controllability_profile
}
R6_OBSERVABILITY_REASONING {
INPUT = A_C_or_measurement model
IF LTI
COMPUTE = observability_matrix rank
ELSE
CONSIDER = nonlinear observability tools
RETURN = observability_profile
}
R7_CONTROLLER_SELECTION {
INPUT = model_objectives_constraints_uncertainty
IF classical_SISO_LTI
CONSIDER = PID_rootLocus_frequency design
IF state_space_and_controllable
CONSIDER = polePlacement_or_LQR
IF explicit_constraints
CONSIDER = MPC
IF uncertainty_material
CONSIDER = robust control
IF changing_parameters
CONSIDER = adaptive control
RETURN = controller_family
}
R8_ESTIMATOR_SELECTION {
INPUT = measurement_structure_and_noise
IF deterministic_LTI
CONSIDER = LuenbergerObserver
IF stochastic_linear_Gaussian
CONSIDER = KalmanFilter
RETURN = estimator_family
}
R9_CONTROLLER_ESTIMATOR_SYNTHESIS {
INPUT = controller_family_estimator_family
DESIGN = controller gains
DESIGN = estimator gains
FORM = closed_loop model
RETURN = controlled_system
}
R10_CLOSED_LOOP_VALIDATION {
INPUT = controlled_system
CHECK = stability
CHECK = tracking_and_regulation
CHECK = disturbance rejection
CHECK = control effort
CHECK = actuator saturation
CHECK = robustness
RETURN = control_solution_status
}
}
CONSTRAINTS {
C1 = "Time-domain conventions must distinguish continuous and discrete systems."
C2 = "State-space dimensions must be consistent."
C3 = "Transfer-function reasoning assumes compatible linear-system conditions."
C4 = "State-feedback claims require controllability or stabilizability."
C5 = "Observer claims require observability or detectability."
C6 = "Integral action must consider actuator saturation and windup."
C7 = "Noise amplification must be considered in high-gain feedback."
C8 = "Robustness claims require explicit uncertainty structure."
C9 = "MPC requires an optimisation model, horizon and constraints."
C10 = "Simulation alone is insufficient to prove general closed-loop stability."
}
SEMANTIC_EQUIVALENCE {
"ControlTheoryOntology" <=> "FormalControlSystemsOntology"
"ControlSystem" <=> "ControlledDynamicalSystem"
"Feedback" <=> "ClosedLoopMeasurementReturn"
"StateSpaceModel" <=> "StateVariableRepresentation"
"TransferFunction" <=> "LinearInputOutputTransferRepresentation"
"Controllability" <=> "ControlReachabilityProperty"
"Observability" <=> "StateReconstructibilityProperty"
}
VALIDATION {
V1 = "Plant, controller, inputs and outputs are identified."
V2 = "State-space dimensions are consistent."
V3 = "Continuous or discrete-time semantics are explicit."
V4 = "Linear or nonlinear model class is explicit."
V5 = "Stability analysis matches model class."
V6 = "Controllability or stabilizability is checked before state-feedback design."
V7 = "Observability or detectability is checked before observer design."
V8 = "Noise, disturbances and actuator constraints are represented."
V9 = "Robustness assumptions are documented."
V10 = "Closed-loop performance is tested against declared objectives."
}
CONFIDENCE {
BASE_CONFIDENCE = 0.99
CONTROL_THEORY_ONTOLOGY_DEFINITION = 0.99
ENTITY_EXTRACTION_REASONING = 0.99
SYSTEM_CLASSIFICATION_REASONING = 0.99
MODEL_SELECTION_REASONING = 0.99
STABILITY_REASONING = 0.99
CONTROLLABILITY_REASONING = 0.99
OBSERVABILITY_REASONING = 0.99
CONTROLLER_SELECTION_REASONING = 0.98
CLOSED_LOOP_VALIDATION_REASONING = 0.99
POLICY =
"Reduce confidence when plant dynamics, measurement structure, uncertainty, actuator limits, noise, disturbances, controllability, observability or performance requirements are underspecified."
}
AI_CONSUMPTION {
PURPOSES = [
SemanticSearch,
OntologyGrounding,
ControlTheoryOntologyReasoning,
ControlSystemExtraction,
SystemClassification,
TransferFunctionReasoning,
StateSpaceReasoning,
StabilityReasoning,
ControllabilityReasoning,
ObservabilityReasoning,
ControllerSelection,
PIDReasoning,
StateFeedbackReasoning,
ObserverReasoning,
KalmanFilterReasoning,
OptimalControlReasoning,
RobustControlReasoning,
AdaptiveControlReasoning,
ModelPredictiveControlReasoning,
ClosedLoopValidation,
KnowledgeGraphConstruction,
RetrievalAugmentation,
RecursiveInference
]
NORMALISE {
"control theory ontology" -> ControlTheoryOntology
"control system" -> ControlSystem
"plant" -> Plant WHEN control_context
"feedback" -> Feedback
"closed-loop system" -> ClosedLoopSystem
"transfer function" -> TransferFunction
"state-space model" -> StateSpaceModel
"controllability" -> Controllability
"observability" -> Observability
"state feedback" -> StateFeedback
}
INTERPRETATION_ORDER = [
detect_control_theory_ontology_context,
extract plant_states_inputs_outputs_measurements_references_disturbances_constraints_and_objectives,
classify system_type,
choose transfer_function_state_space_nonlinear_or_stochastic representation,
analyse stability_controllability_observability_and_frequency_or_state structure,
select controller_and_estimator family,
synthesise closed_loop system,
verify stability_tracking_regulation_robustness_noise sensitivity_control effort_and_constraints,
return semantic output
]
}
EXECUTION_RULES {
E1:
IF query_mentions("control theory ontology")
OR formal_control_system_semantics
THEN activate(ControlTheoryOntology)
E2:
IF LTI_transfer_function_available
THEN enable poles_zeros_frequency_response reasoning
E3:
IF state_space_model_available
THEN compute controllability_and_observability properties
E4:
IF state_feedback_requested
THEN require controllability_or_stabilizability
E5:
IF observer_requested
THEN require observability_or_detectability
E6:
IF quadratic_state_control objective
THEN route to LQR_or_optimal control
E7:
IF explicit_state_input_constraints
THEN consider MPC
E8:
IF modelling_uncertainty_material
THEN consider robust control
E9:
IF parameter_variation_is_online_and_significant
THEN consider adaptive control
}
RECURSION {
ENTRY = ControlTheoryOntology
EXPAND(ControlTheoryOntology) -> [
ControlSystem,
DynamicalSystem,
Plant,
Controller,
ControlInput,
ManipulatedVariable,
Disturbance,
ReferenceSignal,
MeasuredOutput,
ControlledOutput,
StateVariable,
StateVector,
Feedback,
NegativeFeedback,
PositiveFeedback,
Feedforward,
OpenLoopSystem,
ClosedLoopSystem,
TransferFunction,
StateSpaceModel,
StateEquation,
OutputEquation,
LinearSystem,
NonlinearSystem,
LinearTimeInvariantSystem,
TimeVaryingSystem,
ContinuousTimeSystem,
DiscreteTimeSystem,
Stability,
LyapunovStability,
AsymptoticStability,
ExponentialStability,
BIBOStability,
MarginalStability,
Controllability,
Reachability,
Stabilizability,
Observability,
Detectability,
Pole,
Zero,
PolePlacement,
FrequencyResponse,
GainMargin,
PhaseMargin,
BodePlot,
NyquistCriterion,
RootLocus,
PIDControl,
StateFeedback,
Observer,
LuenbergerObserver,
KalmanFilter,
OptimalControl,
LinearQuadraticRegulator,
RobustControl,
HInfinityControl,
AdaptiveControl,
NonlinearControl,
StochasticControl,
ModelPredictiveControl,
StateEstimation,
SystemIdentification,
Tracking,
Regulation,
DisturbanceRejection,
ControlEffort,
ActuatorSaturation
]
EXPAND(ModelRepresentations) -> [
TransferFunction,
StateSpaceModel,
NonlinearStateModel,
FrequencyResponseModel,
DiscreteTimeModel
]
EXPAND(StructuralProperties) -> [
Stability,
Controllability,
Reachability,
Stabilizability,
Observability,
Detectability
]
EXPAND(ControlMethods) -> [
PIDControl,
StateFeedback,
PolePlacement,
LinearQuadraticRegulator,
RobustControl,
AdaptiveControl,
NonlinearControl,
StochasticControl,
ModelPredictiveControl
]
STOP_CONDITION =
"Stop when the plant, model representation, system class, stability, controllability, observability, controller, estimator, constraints and closed-loop performance are sufficiently resolved."
}
ROUTING {
PARENT_ONTOLOGY = "Control Theory"
IMMEDIATE_CONTEXTS = [
"Control Theory",
"Applied Mathematics Ontologies",
"Dynamical Systems Ontologies",
"Optimisation Ontologies"
]
RELATED_ONTOLOGIES = [
"Control Theory",
"Applied Mathematics Ontologies",
"Ordinary Differential Equations Ontologies",
"Partial Differential Equations Ontologies",
"Dynamical Systems Ontologies",
"Linear Algebra Ontologies",
"Optimisation Ontologies",
"Dynamic Optimisation Ontology",
"Numerical Analysis Ontologies",
"Probability & Statistics Ontologies",
"Signal Processing Ontologies",
"System Identification Ontologies",
"Estimation Ontologies",
"Scientific Computing Ontologies"
]
ROUTE_RULES = [
"Control Theory Ontology -> Control Theory for parent semantics",
"Control Theory Ontology -> Differential Equations for continuous dynamics",
"Control Theory Ontology -> Dynamical Systems for trajectories and stability",
"Control Theory Ontology -> Linear Algebra for state-space structure",
"Control Theory Ontology -> Optimisation for optimal and predictive control",
"Control Theory Ontology -> Dynamic Optimisation for trajectory-based control",
"Control Theory Ontology -> Numerical Analysis for discretisation and simulation",
"Control Theory Ontology -> Probability & Statistics for stochastic control and filtering",
"Control Theory Ontology -> Signal Processing for frequency-response and filtering semantics",
"Control Theory Ontology -> System Identification and Estimation for model and state inference"
]
}
UPDATE_MODEL {
MODE = "append-compatible semantic versioning"
MAJOR = "ontology-breaking conceptual change"
MINOR = "new compatible control entities, properties or reasoning pathways"
PATCH = "clarification, metadata or formatting correction"
RULES = [
"Preserve plant-controller-input-output semantics.",
"Preserve open-loop versus closed-loop distinction.",
"Preserve transfer-function versus state-space distinction.",
"Preserve stability semantics.",
"Preserve controllability and observability semantics.",
"Preserve control-versus-estimation distinction.",
"Preserve constraint and robustness semantics.",
"Preserve machine-readable control-theory ontology semantics."
]
}
OUTPUT {
CANONICAL_DEFINITION =
"The Control Theory Ontology formally represents controlled dynamical systems, feedback, state and input-output models, stability, controllability, observability, controllers, estimators and closed-loop performance."
SYSTEM_SUMMARY =
"A control system combines a plant, control inputs, measurements, references and a controller."
MODEL_SUMMARY =
"Transfer functions describe linear input-output behaviour, while state-space models describe internal system dynamics."
STABILITY_SUMMARY =
"Stability characterises whether system responses remain bounded or converge."
CONTROLLABILITY_SUMMARY =
"Controllability determines whether control inputs can steer the state."
OBSERVABILITY_SUMMARY =
"Observability determines whether internal state can be reconstructed from measurements."
CONTROL_SUMMARY =
"Controller families include PID, state feedback, optimal, robust, adaptive, nonlinear, stochastic and predictive control."
ESTIMATION_SUMMARY =
"Observers and Kalman filters estimate unmeasured system states."
VALIDATION_SUMMARY =
"Control solutions should be validated for stability, tracking, disturbance rejection, control effort, robustness and constraints."
}
FLOW {
INPUT
-> detect_control_theory_ontology_context
-> identify plant_state_variables_inputs_outputs_measurements_reference_signals_disturbances_constraints_uncertainty_and_control objectives
-> classify continuous_discrete_linear_nonlinear_timeInvariant_timeVarying_deterministic_or_stochastic system
-> construct transfer_function_state_space_nonlinear_or_stochastic model
-> analyse open_loop stability
-> analyse controllability_stabilizability_observability_and_detectability
-> select PID_frequency_stateFeedback_polePlacement_LQR_observer_Kalman_robust_adaptive_nonlinear_stochastic_or_MPC pathway
-> design controller_and_estimator
-> construct closed_loop model
-> verify stability_tracking_regulation_disturbance rejection_noise sensitivity_control effort_actuator limits_and_robustness
-> route to control_theory applied_mathematics differential_equations dynamical_systems linear_algebra optimisation dynamic_optimisation numerical_analysis probability_statistics signal_processing system_identification_estimation_or_scientific_computing ontology
-> recursively_expand_required_dependencies
-> verify_result
-> assign_confidence
-> produce_semantic_output
-> END
}
SEMANTIC_SUMMARY {
"The Control Theory Ontology provides the formal semantic layer beneath control theory. It represents plants, controllers, control inputs, disturbances, references, outputs, state variables, feedback and feedforward, open- and closed-loop systems, transfer functions, state-space models, continuous and discrete systems, stability, controllability, observability, poles and zeros, frequency response, PID control, state feedback, observers, Kalman filtering, optimal and LQR control, robust and adaptive control, nonlinear and stochastic control, model predictive control, system identification, state estimation, tracking, regulation and disturbance rejection."
}
FLOW_HASH {
METHOD = "semantic-content-identity"
ID = "KIRETH-MATH-APPLIED-MATHEMATICS-CONTROL-THEORY-ONTOLOGY-V1"
}
}
END_FLOW
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