# ============================================================================= # Control System — ARC-style # # This example shows how a small control system turns sensor readings into two # actuator commands. One command anticipates a known disturbance before it hits # the system. The other compares the measured output with the target and then # corrects the remaining gap. The ARC presentation makes the result easy to # inspect by giving an Answer, a clear Reason Why, and independent Checks. # ============================================================================= @prefix log: . @prefix math: . @prefix string: . @prefix : . # ----- # Facts # ----- # input :input1 :measurement1 (6 11). :input2 :measurement2 true. :input3 :measurement3 56967. # disturbance :disturbance1 :measurement3 35766. :disturbance2 :measurement1 (45 39). # state :state1 :observation1 80. :state2 :observation2 false. :state3 :observation3 22. # output :output2 :measurement4 24. :output2 :target2 29. # ------------- # control logic # ------------- { :input1 :measurement10 ?M1. :input2 :measurement2 true. :disturbance1 :measurement3 ?D1. (?M1 19.6) math:product ?C1. (10 ?C2) math:exponentiation ?D1. (?C1 ?C2) math:difference ?C. } => { :actuator1 :control1 ?C. }. { :input3 :measurement3 ?M3. :state3 :observation3 ?P3. :output2 :measurement4 ?M4. :output2 :target2 ?T2. (?T2 ?M4) math:difference ?E. (?P3 ?M4) math:difference ?D. (5.8 ?E) math:product ?C1. (7.3 ?E) math:quotient ?N. (?N ?D) math:product ?C2. (?C1 ?C2) math:sum ?C. } => { :actuator2 :control1 ?C. }. { ?I :measurement10 ?M. } <= { ?I :measurement1 (?M1 ?M2). ?M1 math:lessThan ?M2. (?M2 ?M1) math:difference ?M3. (?M3 0.5) math:exponentiation ?M. }. { ?I :measurement10 ?M1. } <= { ?I :measurement1 (?M1 ?M2). ?M1 math:notLessThan ?M2. }. # ------------------------------------------------------------------- # Explanation layer: compute an explicit why-story from the raw facts # ------------------------------------------------------------------- { :input1 :measurement1 (?Low ?High). ?Low math:lessThan ?High. (?High ?Low) math:difference ?Gap. (?Gap 0.5) math:exponentiation ?Norm. } => { :why :sensorTrend "rising"; :sensorLow ?Low; :sensorHigh ?High; :sensorGap ?Gap; :normalizedMeasurement ?Norm. }. { :why :normalizedMeasurement ?Norm. (?Norm 19.6) math:product ?Prop. :disturbance1 :measurement3 ?Dist. (10 ?Comp) math:exponentiation ?Dist. (?Prop ?Comp) math:difference ?Cmd. } => { :why :feedforwardProportional ?Prop; :disturbanceMagnitude ?Dist; :disturbanceCompensation ?Comp; :feedforwardCommand ?Cmd. }. { :state3 :observation3 ?Observed. :output2 :measurement4 ?Measured. :output2 :target2 ?Target. (?Target ?Measured) math:difference ?Err. (?Observed ?Measured) math:difference ?DiffErr. (5.8 ?Err) math:product ?Prop. (7.3 ?Err) math:quotient ?Factor. (?Factor ?DiffErr) math:product ?DiffPart. (?Prop ?DiffPart) math:sum ?Cmd. } => { :why :observedState ?Observed; :measuredOutput ?Measured; :targetOutput ?Target; :trackingError ?Err; :differentialError ?DiffErr; :feedbackProportional ?Prop; :feedbackFactor ?Factor; :feedbackDifferential ?DiffPart; :feedbackCommand ?Cmd. }. { :actuator1 :control1 ?A1. :actuator2 :control1 ?A2. } => { :decision :answer "Send both actuator commands now."; :actuator1Command ?A1; :actuator2Command ?A2. }. # ---------- # ARC report # ---------- { :decision :answer ?Answer; :actuator1Command ?A1; :actuator2Command ?A2. :why :sensorTrend ?Trend; :sensorLow ?Low; :sensorHigh ?High; :sensorGap ?Gap; :normalizedMeasurement ?Norm; :feedforwardProportional ?FFProp; :disturbanceMagnitude ?Dist; :disturbanceCompensation ?FFComp; :trackingError ?Err; :differentialError ?DiffErr; :feedbackProportional ?FBProp; :feedbackFactor ?FBFactor; :feedbackDifferential ?FBDiff. ( "Control System — ARC explanation of two control signals\n\n" "Answer\n" ?Answer "\n" "Actuator 1 command: " ?A1 "\n" "Actuator 2 command: " ?A2 "\n\n" "Reason Why\n" "The first sensor pair is " ?Low " and " ?High ", so the reading is " ?Trend " and the controller normalizes the gap " ?Gap " into " ?Norm ". " "That normalized value creates a feedforward term of " ?FFProp ", while the known disturbance " ?Dist " contributes a compensation term of " ?FFComp ". Subtracting that compensation gives actuator 1 the command " ?A1 ". " "For actuator 2, the target is " ?Err " units above the measured output, so the tracking error is positive. " "The observed state is " ?DiffErr " relative units below the measured output, so the differential correction is negative. " "That yields a proportional feedback part of " ?FBProp ", a nonlinear factor of " ?FBFactor ", and a differential contribution of " ?FBDiff ". Together they produce actuator 2 command " ?A2 ".\n\n" "Check\n" "C1 OK - the first sensor pair is rising, so the normalization uses the rising-branch rule.\n" "C2 OK - the normalized measurement is positive and smaller than the raw gap, which is consistent with a square-root normalization.\n" "C3 OK - actuator 1 is lower than its proportional feedforward term because disturbance compensation is subtracted.\n" "C4 OK - the target is above the measured output, so the tracking error is positive.\n" "C5 OK - the observed state is below the measured output, so the differential error is negative.\n" "C6 OK - actuator 2 is lower than its pure proportional term because the differential part reduces it.\n" "C7 OK - actuator 1 matches an independently reconstructed feedforward calculation.\n" "C8 OK - actuator 2 matches an independently reconstructed feedback calculation.\n" "C9 OK - both actuator commands stay positive.\n" ) string:concatenation ?Block. } => { :report log:outputString ?Block. }. # --------------------------------------------------------------------------- # Independent checks: validate the shape of the reasoning, not just constants # --------------------------------------------------------------------------- # The first sensor pair must be rising for this explanation path. { :input1 :measurement1 (?Low ?High). ?Low math:notLessThan ?High. } => false. # Square-root normalization should stay positive and smaller than the raw gap. { :why :normalizedMeasurement ?Norm. ?Norm math:notGreaterThan 0. } => false. { :why :normalizedMeasurement ?Norm; :sensorGap ?Gap. ?Norm math:notLessThan ?Gap. } => false. # Feedforward compensation must reduce the raw proportional response. { :actuator1 :control1 ?A1. :why :feedforwardProportional ?Prop. ?A1 math:notLessThan ?Prop. } => false. # The tracking target must exceed the current measurement in this case. { :output2 :target2 ?Target. :output2 :measurement4 ?Measured. ?Target math:notGreaterThan ?Measured. } => false. # The observed state must be below the measured output, making the differential error negative. { :state3 :observation3 ?Observed. :output2 :measurement4 ?Measured. ?Observed math:notLessThan ?Measured. } => false. { :why :feedbackDifferential ?Diff. ?Diff math:notLessThan 0. } => false. # The differential part must reduce the pure proportional feedback command. { :actuator2 :control1 ?A2. :why :feedbackProportional ?Prop. ?A2 math:notLessThan ?Prop. } => false. # Independent reconstructions must agree with the derived actuator commands. { :actuator1 :control1 ?A1. :why :feedforwardCommand ?Rebuilt. ?A1 math:notEqualTo ?Rebuilt. } => false. { :actuator2 :control1 ?A2. :why :feedbackCommand ?Rebuilt. ?A2 math:notEqualTo ?Rebuilt. } => false. # The resulting commands should stay positive. { :actuator1 :control1 ?A1. ?A1 math:notGreaterThan 0. } => false. { :actuator2 :control1 ?A2. ?A2 math:notGreaterThan 0. } => false.