Control Theory

AI INTERPRETATION NOTE
This Flow Algebra defines the Control Theory layer within the Kireth Research Semantic Library. Control theory studies how dynamical systems can be analysed and influenced through feedback, feedforward, estimation and optimisation so that desired behaviour is achieved despite disturbances, uncertainty and constraints. This parent ontology connects classical control, modern control, state-space methods, stability, controllability, observability, feedback, frequency response, optimal control, robust control, adaptive control, nonlinear control, stochastic control, model predictive control, system identification and estimation. It provides the semantic routing layer between applied mathematics, differential equations, optimisation, signal processing, dynamical systems and engineering.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = ControlTheory
 SYMBOLIC_FORMS = [
  "x_dot = A x + B u",
  "y = C x + D u",
  "u = -Kx",
  "G(s)=Y(s)/U(s)"
 ]
 ENTITY_CLASS = DynamicalSystemRegulationFramework
 PRIMARY_CONTEXT = AppliedMathematics
 DEFINITION = "Control theory is the mathematical study of dynamical systems, feedback, estimation and control laws used to achieve desired system behaviour."
 SEMANTIC_ROLE = "Provides the parent semantic framework for system dynamics, input-output relations, feedback, stability, controllability, observability, frequency-domain analysis, optimal control, robust control, estimation and constrained control."
}

SCOPE {
 INCLUDE = [
  ControlTheory,
  DynamicalSystem,
  Plant,
  Controller,
  ControlInput,
  Disturbance,
  ReferenceSignal,
  Output,
  StateVariable,
  Feedback,
  Feedforward,
  OpenLoopSystem,
  ClosedLoopSystem,
  TransferFunction,
  StateSpaceModel,
  LinearTimeInvariantSystem,
  NonlinearSystem,
  ContinuousTimeSystem,
  DiscreteTimeSystem,
  Stability,
  LyapunovStability,
  AsymptoticStability,
  BIBOStability,
  Controllability,
  Observability,
  Reachability,
  Detectability,
  Stabilizability,
  Pole,
  Zero,
  PolePlacement,
  FrequencyResponse,
  BodePlot,
  NyquistCriterion,
  RootLocus,
  PIDControl,
  StateFeedback,
  Observer,
  KalmanFilter,
  OptimalControl,
  LinearQuadraticRegulator,
  RobustControl,
  HInfinityControl,
  AdaptiveControl,
  NonlinearControl,
  StochasticControl,
  ModelPredictiveControl,
  SystemIdentification,
  StateEstimation,
  DisturbanceRejection,
  Tracking,
  Regulation
 ]

 EXCLUDE_AS_PRIMARY = [
  PureDifferentialEquationAnalysisWithoutControlInput,
  SignalProcessingWithoutFeedbackOrControlContext,
  OptimisationWithoutDynamicalSystemContext,
  MechanicalDesignWithoutControlLawSemantics
 ]

 DISAMBIGUATION_RULE =
  "When a dynamical system is analysed or influenced through inputs, feedback, estimation or control laws to achieve desired behaviour, interpret the context as control theory."
}

CORE_CONCEPTS {

 ControlTheory {
  definition = "The mathematical discipline concerned with analysis and regulation of dynamical systems."
 }

 DynamicalSystem {
  definition = "A system whose state evolves over time according to differential, difference or other evolution laws."
 }

 Plant {
  definition = "The system or process to be controlled."
 }

 Controller {
  definition = "A mechanism or law that computes control actions from references, measurements or state estimates."
 }

 Feedback {
  definition = "The use of measured or estimated system output or state to influence future control actions."
 }

 OpenLoopSystem {
  definition = "A system in which control action does not depend on measured output."
 }

 ClosedLoopSystem {
  definition = "A system in which feedback influences the applied control."
 }

 StateSpaceModel {
  definition = "A dynamical representation using state variables and first-order evolution equations."
 }

 TransferFunction {
  definition = "A Laplace-domain or z-domain input-output representation of a linear time-invariant system under zero initial conditions."
 }

 Stability {
  definition = "A property describing whether system behaviour remains bounded or converges after perturbation."
 }

 Controllability {
  definition = "The ability to steer a system state using admissible control inputs."
 }

 Observability {
  definition = "The ability to reconstruct the internal state from measured outputs over time."
 }
}

RELATIONS {
 ControlTheory --is_branch_of--> AppliedMathematics
 ControlTheory --uses--> DynamicalSystems
 ControlTheory --uses--> DifferentialEquations
 ControlTheory --uses--> LinearAlgebra
 ControlTheory --uses--> Optimisation
 Plant --is_controlled_by--> Controller
 Controller --produces--> ControlInput
 ControlInput --acts_on--> Plant
 Disturbance --acts_on--> Plant
 Plant --produces--> Output
 Feedback --uses--> Output
 Feedback --influences--> Controller
 StateSpaceModel --represents--> DynamicalSystem
 TransferFunction --represents--> LinearTimeInvariantSystem
 Pole --determines--> LinearStability
 Controllability --supports--> StateFeedbackDesign
 Observability --supports--> StateEstimation
 Observer --estimates--> StateVariable
 KalmanFilter --is_type_of--> StateEstimator
 OptimalControl --uses--> Optimisation
 ModelPredictiveControl --uses--> RecedingHorizonOptimisation
 RobustControl --addresses--> ModelUncertainty
 AdaptiveControl --adjusts_to--> ChangingSystemParameters
}

OPERATORS {
 STATE_DYNAMICS_CONTINUOUS(A,B,x,u) :=
  x_dot=A*x+B*u

 OUTPUT_EQUATION(C,D,x,u) :=
  y=C*x+D*u

 STATE_DYNAMICS_DISCRETE(A,B,x_k,u_k) :=
  x_{k+1}=A*x_k+B*u_k

 STATE_FEEDBACK(K,x) :=
  u=-K*x

 CLOSED_LOOP_MATRIX(A,B,K) :=
  A-B*K

 TRANSFER_FUNCTION(A,B,C,D,s) :=
  C*(sI-A)^(-1)*B + D

 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

 LYAPUNOV_TEST(A,P,Q) :=
  A^T*P+P*A=-Q
  with P>0 and Q>0

 LQR_COST(x,u,Q,R) :=
  integral (x^T Q x + u^T R u) dt

 PID_CONTROL(e,Kp,Ki,Kd) :=
  Kp*e + Ki*integral(e dt) + Kd*de/dt

 TRACKING_ERROR(r,y) :=
  e=r-y
}

DEFINITIONS {
 D1 = "Control theory studies how inputs influence dynamical systems."
 D2 = "Feedback uses measured or estimated system behaviour to modify control action."
 D3 = "Open-loop and closed-loop systems differ by the presence of feedback."
 D4 = "State-space models represent internal dynamic state."
 D5 = "Transfer functions provide input-output descriptions of linear time-invariant systems."
 D6 = "Stability is a fundamental prerequisite for many control objectives."
 D7 = "Controllability concerns whether states can be reached by control."
 D8 = "Observability concerns whether states can be inferred from outputs."
 D9 = "Optimal control combines dynamics with performance objectives."
 D10 = "Robust control addresses modelling uncertainty and disturbances."
}

AXIOMS {
 A1_DYNAMICS:
  ControlTheory operates on systems with explicit state or input-output dynamics

 A2_CONTROL_INPUT:
  Controlled behaviour depends on one or more manipulable inputs

 A3_FEEDBACK:
  Feedback control uses measured or estimated system information

 A4_STABILITY:
  Closed-loop stability is a primary control-design requirement

 A5_CONTROLLABILITY:
  State-feedback design requires appropriate controllability or stabilizability properties

 A6_OBSERVABILITY:
  State reconstruction requires observability or detectability properties

 A7_MODEL_DEPENDENCE:
  Controller design depends on system models, identified data or robust uncertainty descriptions

 A8_PERFORMANCE:
  Control quality is evaluated through tracking, regulation, disturbance rejection, robustness, effort or other declared criteria
}

REASONING {

 R1_SYSTEM_IDENTIFICATION {
  INPUT = control_problem
  IDENTIFY = plant_states_inputs_outputs_disturbances_and_reference signals
  RETURN = control_system_structure
 }

 R2_MODEL_CLASSIFICATION {
  INPUT = system_structure
  CLASSIFY = continuous_or_discrete
  CLASSIFY = linear_or_nonlinear
  CLASSIFY = deterministic_or_stochastic
  CLASSIFY = time_invariant_or_time_varying
  RETURN = system_class
 }

 R3_STABILITY_ANALYSIS {
  INPUT = system_model
  IF LTI_continuous
   ANALYSE = eigenvalues_or_poles
  IF nonlinear
   CONSIDER = Lyapunov_methods
  RETURN = stability_profile
 }

 R4_CONTROLLABILITY_ANALYSIS {
  INPUT = A_B_or_general_dynamics
  COMPUTE = controllability_or_reachability properties
  RETURN = controllability_profile
 }

 R5_OBSERVABILITY_ANALYSIS {
  INPUT = A_C_or_measurement_model
  COMPUTE = observability_or_detectability properties
  RETURN = observability_profile
 }

 R6_CLASSICAL_CONTROL {
  INPUT = transfer_function_and_performance targets
  ANALYSE = poles_zeros_frequency_response_root locus
  DESIGN = PID_lead_lag_or_frequency compensator
  RETURN = classical_controller
 }

 R7_STATE_SPACE_CONTROL {
  INPUT = state_space_model
  IF controllable
   DESIGN = pole_placement_or_LQR
  IF state_not_measured
   DESIGN = observer_or_Kalman filter
  RETURN = state_space_controller
 }

 R8_ROBUST_AND_ADAPTIVE_CONTROL {
  INPUT = model_uncertainty_or_parameter variation
  SELECT = robust_control_adaptive_control_or_gain scheduling
  RETURN = uncertainty_aware controller
 }

 R9_OPTIMAL_AND_PREDICTIVE_CONTROL {
  INPUT = dynamics_cost_constraints
  FORM = optimal_control_problem
  IF receding_horizon
   SOLVE = model_predictive_control problem
  RETURN = optimal_control law_or_policy
 }

 R10_CLOSED_LOOP_VALIDATION {
  INPUT = designed_controller
  CHECK = stability_tracking_regulation_disturbance_rejection_control effort_and_constraints
  RETURN = control_performance profile
 }
}

CONSTRAINTS {
 C1 = "Continuous-time and discrete-time stability criteria differ."
 C2 = "Transfer-function methods assume compatible linear-system conditions."
 C3 = "Pole placement requires appropriate controllability."
 C4 = "Observer design requires appropriate observability or detectability."
 C5 = "High controller gain can amplify noise or excite unmodelled dynamics."
 C6 = "Integral action can cause windup under actuator saturation."
 C7 = "Model predictive control requires explicit constraints and repeated optimisation."
 C8 = "Robustness claims require a declared uncertainty model."
 C9 = "Adaptive control requires identifiable or sufficiently exciting system behaviour under relevant formulations."
 C10 = "Simulation success alone does not prove stability or robustness."
}

SEMANTIC_EQUIVALENCE {
 "ControlTheory" <=> "AutomaticControlTheory"
 "ClosedLoopSystem" <=> "FeedbackControlledSystem"
 "StateSpaceModel" <=> "StateVariableModel"
 "TransferFunction" <=> "InputOutputTransferRepresentation"
 "Controllability" <=> "StateReachabilityUnderControl"
 "Observability" <=> "StateReconstructibilityFromOutput"
}

VALIDATION {
 V1 = "Plant, inputs and outputs are identified."
 V2 = "Continuous or discrete time is explicit."
 V3 = "Linear or nonlinear model class is explicit."
 V4 = "Stability is analysed using compatible criteria."
 V5 = "Controllability or stabilizability is checked before state-feedback claims."
 V6 = "Observability or detectability is checked before observer claims."
 V7 = "Controller limits and actuator constraints are represented."
 V8 = "Noise and disturbance assumptions are documented."
 V9 = "Robustness claims match declared uncertainty."
 V10 = "Closed-loop performance is verified against stated objectives."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 CONTROL_THEORY_DEFINITION = 0.99
 SYSTEM_CLASSIFICATION_REASONING = 0.99
 STABILITY_REASONING = 0.99
 CONTROLLABILITY_REASONING = 0.99
 OBSERVABILITY_REASONING = 0.99
 CLASSICAL_CONTROL_REASONING = 0.98
 STATE_SPACE_REASONING = 0.99
 ROBUST_CONTROL_REASONING = 0.97
 OPTIMAL_CONTROL_REASONING = 0.98

 POLICY =
  "Reduce confidence when system dynamics, time domain, measurement structure, uncertainty, actuator limits, disturbance model, controllability, observability or performance objectives are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  ControlTheoryReasoning,
  DynamicalSystemReasoning,
  FeedbackReasoning,
  StabilityReasoning,
  ControllabilityReasoning,
  ObservabilityReasoning,
  TransferFunctionReasoning,
  StateSpaceReasoning,
  FrequencyResponseReasoning,
  PIDControlReasoning,
  StateFeedbackReasoning,
  ObserverReasoning,
  KalmanFilterReasoning,
  OptimalControlReasoning,
  RobustControlReasoning,
  AdaptiveControlReasoning,
  ModelPredictiveControlReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "control theory" -> ControlTheory
  "automatic control" -> ControlTheory
  "feedback control" -> Feedback
  "closed-loop control" -> ClosedLoopSystem
  "state-space model" -> StateSpaceModel
  "transfer function" -> TransferFunction
  "controllability" -> Controllability
  "observability" -> Observability
  "PID controller" -> PIDControl
  "model predictive control" -> ModelPredictiveControl
 }

 INTERPRETATION_ORDER = [
  detect_control_theory_context,
  identify plant_states_inputs_outputs_references_disturbances_measurements_and_constraints,
  classify continuous_discrete_linear_nonlinear_deterministic_stochastic_and_time_variation,
  analyse stability_controllability_observability_and_frequency_or_state structure,
  select classical_state_space_optimal_robust_adaptive_nonlinear_stochastic_or_predictive control pathway,
  design_or_assess controller_and_estimator,
  verify closed_loop stability_tracking_regulation_robustness_control effort_and_constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("control theory")
  OR dynamical_system_with_control_input_and_feedback_context
  THEN activate(ControlTheoryOntology)

 E2:
  IF transfer_function_available_and_system_LTI
  THEN consider classical_frequency_domain methods

 E3:
  IF state_space_model_available
  THEN assess controllability_and_observability

 E4:
  IF desired_closed_loop poles specified
  THEN consider pole_placement_when controllable

 E5:
  IF quadratic_state_and_control cost
  THEN consider LQR

 E6:
  IF measurement_noise_and_process_noise present
  THEN consider Kalman filtering

 E7:
  IF explicit state_input constraints matter
  THEN consider model_predictive_control

 E8:
  IF model_uncertainty_material
  THEN consider robust_control

 E9:
  IF plant_parameters_change_online
  THEN consider adaptive_control_or_gain scheduling
}

RECURSION {
 ENTRY = ControlTheory

 EXPAND(ControlTheory) -> [
  DynamicalSystem,
  Plant,
  Controller,
  ControlInput,
  Disturbance,
  ReferenceSignal,
  Output,
  StateVariable,
  Feedback,
  Feedforward,
  OpenLoopSystem,
  ClosedLoopSystem,
  TransferFunction,
  StateSpaceModel,
  LinearTimeInvariantSystem,
  NonlinearSystem,
  ContinuousTimeSystem,
  DiscreteTimeSystem,
  Stability,
  LyapunovStability,
  AsymptoticStability,
  BIBOStability,
  Controllability,
  Observability,
  Reachability,
  Detectability,
  Stabilizability,
  Pole,
  Zero,
  PolePlacement,
  FrequencyResponse,
  BodePlot,
  NyquistCriterion,
  RootLocus,
  PIDControl,
  StateFeedback,
  Observer,
  KalmanFilter,
  OptimalControl,
  LinearQuadraticRegulator,
  RobustControl,
  HInfinityControl,
  AdaptiveControl,
  NonlinearControl,
  StochasticControl,
  ModelPredictiveControl,
  SystemIdentification,
  StateEstimation,
  DisturbanceRejection,
  Tracking,
  Regulation
 ]

 EXPAND(ClassicalControl) -> [
  TransferFunction,
  Pole,
  Zero,
  FrequencyResponse,
  BodePlot,
  NyquistCriterion,
  RootLocus,
  PIDControl,
  LeadCompensator,
  LagCompensator
 ]

 EXPAND(StateSpaceControl) -> [
  StateSpaceModel,
  Controllability,
  Observability,
  PolePlacement,
  StateFeedback,
  Observer,
  KalmanFilter,
  LinearQuadraticRegulator
 ]

 EXPAND(AdvancedControl) -> [
  RobustControl,
  HInfinityControl,
  AdaptiveControl,
  NonlinearControl,
  StochasticControl,
  ModelPredictiveControl,
  OptimalControl
 ]

 STOP_CONDITION =
  "Stop when the plant model, system class, stability, controllability, observability, controller structure, estimator, robustness, constraints and closed-loop performance are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Applied Mathematics Ontologies"

 IMMEDIATE_CONTEXTS = [
  "Applied Mathematics Ontologies",
  "Differential Equations Ontologies",
  "Dynamical Systems Ontologies",
  "Optimisation Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "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 -> Applied Mathematics Ontologies for parent semantics",
  "Control Theory -> Differential Equations for continuous system dynamics",
  "Control Theory -> Dynamical Systems for trajectories and stability",
  "Control Theory -> Linear Algebra for state-space structure",
  "Control Theory -> Optimisation for optimal and predictive control",
  "Control Theory -> Dynamic Optimisation for trajectory control problems",
  "Control Theory -> Numerical Analysis for simulation and discretisation",
  "Control Theory -> Probability & Statistics for stochastic systems and filtering",
  "Control Theory -> Signal Processing for frequency-domain and filtering methods",
  "Control Theory -> System Identification for model construction from data"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible control classes, stability notions or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve plant-controller-feedback semantics.",
  "Preserve continuous-versus-discrete distinction.",
  "Preserve state-space and transfer-function distinctions.",
  "Preserve stability semantics.",
  "Preserve controllability and observability semantics.",
  "Preserve estimation-versus-control distinction.",
  "Preserve robust and constrained control semantics.",
  "Preserve machine-readable control-theory semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Control theory studies how dynamical systems can be analysed and influenced through control inputs, feedback and estimation to achieve desired behaviour."

 FEEDBACK_SUMMARY =
  "Feedback uses measured or estimated system behaviour to modify future control actions."

 STABILITY_SUMMARY =
  "Stability determines whether closed-loop responses remain bounded or converge after perturbations."

 CONTROLLABILITY_SUMMARY =
  "Controllability describes whether admissible inputs can steer the system state."

 OBSERVABILITY_SUMMARY =
  "Observability describes whether internal state can be reconstructed from outputs."

 CLASSICAL_SUMMARY =
  "Classical control uses transfer functions, poles, zeros and frequency response."

 STATE_SPACE_SUMMARY =
  "Modern state-space control uses matrix models, state feedback and observers."

 OPTIMAL_SUMMARY =
  "Optimal control chooses inputs to minimise a performance criterion subject to system dynamics."

 ADVANCED_SUMMARY =
  "Robust, adaptive, nonlinear, stochastic and predictive control address uncertainty, nonlinearities, changing systems and explicit constraints."
}

FLOW {
 INPUT
 -> detect_control_theory_context
 -> identify plant_states_inputs_outputs_measurements_references_disturbances_constraints_and_objectives
 -> classify continuous_discrete_linear_nonlinear_timeInvariant_timeVarying_deterministic_or_stochastic system
 -> construct transfer_function_state_space_or_nonlinear dynamic model
 -> analyse open_loop_and_closed_loop stability
 -> assess controllability_stabilizability_observability_and_detectability
 -> select classical_PID_frequency stateFeedback_observer_LQR_robust_adaptive_nonlinear_stochastic_or_modelPredictive control pathway
 -> design_or_assess controller_and_estimator
 -> evaluate tracking_regulation_disturbance rejection_noise sensitivity_control effort_constraints_and_robustness
 -> route to 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 layer defines mathematical analysis and regulation of dynamical systems. It supports plants, controllers, inputs, outputs, disturbances, references, feedback and feedforward, transfer functions, state-space models, continuous and discrete systems, stability, controllability, observability, poles and zeros, frequency response, Bode and Nyquist methods, root locus, PID control, state feedback, observers, Kalman filtering, optimal and LQR control, robust and H-infinity control, adaptive and nonlinear control, 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-V1"
}

}

END_FLOW