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The Morven Framework

The Morven Framework. George M. Coghill. Motivation. To provide properly constructive, constraint based qualitative simulation Retain QR ethos To alleviate the problem of spurious behaviours General purpose QR Why a “Framework” No system is suitable for all situations

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The Morven Framework

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  1. The Morven Framework George M. Coghill

  2. Motivation • To provide properly constructive, constraint based qualitative simulation • Retain QR ethos • To alleviate the problem of spurious behaviours • General purpose QR • Why a “Framework” • No system is suitable for all situations • Permits testing and comparison of approaches • Consists in modular constituents

  3. Context Qualitative TQA & TCP V.E. P.A. Morven Predictive Vector Envisionment Algorithm FuSim QSIM Reasoning

  4. Constituents • Predecessors • Variables are represented as vectors • Models are distributed over differential planes • Fuzzy quantity spaces are utilised • Empirical knowledge can be incorporated. • Specific to Morven • Transitions only generated for state variables • Constructive (assynchronous) simulation • Fuzzy Vector Envisionment • Different approach to prioritisation • Discrete time (synchronous) simulation

  5. Constiuents (2) • Permits multi-dimensional comparisons • Constructive & Non-constructive • Simulation & Envisionment • Synchronous & Assynchronous

  6. Simulation Synchronous Non-constructive Constructive Asynchronous Envisionment The Morven Framework

  7. Fuzzy Qualitative Reasoning • Motivation • Integration of qualitative and vague quantitative information - captured in the nature of fuzzy sets • Ability to utilise and calculate temporal information in a qualitative simulator • To include empirically derived information into a qualitative simulator

  8. 1 1 x 0 0 x b a a ( a ) ( b )  (x)  (x) A A 1 1 0 0 x a a b x ( c ) ( d )  (x)  (x) A A  b+ a- a+  a-   A convenient fuzzy representation • 4-tuple fuzzy numbers (a, b, ) • precise and approximate • useful for computation

  9. (x) m A Fuzzy Quantity Spaces 0 x -0.4 0.6 -1 -0.8 -0.6 -0.2 0 0.2 0.4 0.8 1

  10. _ + 0 + 0 _ Curve Shapes • d2 • d1

  11. [++] [+o] [+-] [o+] [oo] [o-] [-+] [-o] [- -] Transition Rules • Intermediate Value Theorem (IVT) • States that for a continuous system, a function joining two points of opposite sign must pass through zero. • Mean Value Theorem (MVT) • Defines the direction of change of a variable between two points.

  12. Single Tank System plane 0 qO = kV V’ = qi - qO plane 1 q’O = kV’ V’’ = q’i - q’O plane 2 q’’O = kV’’ V’’’ = q’’i - q’’O qi V qo

  13. u 1 k10.x1 Single Compartment System plane 0 k10x1 = k10.x1 x1’ = u - k10x1 plane 1 k10x1’ = k10.x1’ x1’’ = u’ - k10x1’ plane 2 k10x1’’ = k10.x1’’ x1’’’ = u’’ - k10x1’’

  14. Models in Morven (define-fuzzy-model<model_name> (short-name<short_name_of_model>) (variables<list-of [variable_name, bounds, quantity-space]>) (auxiliary-variables<list-of auxiliary_variable_names>) (input <list-of [input_name, bounds, quantity-space]>) (constraints<list-of [differential_planes (list-of constraints)]> (print<list-of variable_names>) )

  15. A JMorven Model model-name: single-tank short-name: fst NumSystemVariables: 2 variable: qo range: zero p-max NumDerivatives: 1 qspaces: tanks-quantity-space variable: V range: zero p-max NumDerivatives: 2 qsapces: tanks-quantity-space tanks-quantity-space2 NumExogenousVariables: 1 variable: qi range: zero p-max NumDerivatives: 1 qspaces: tanks-quantity-space Constraints: NumDiffPlanes: 2 Plane: 0 NumConstraints: 2 Constraint: func (dt 0 qo) (dt 0 V) NumMappings: 9 Mappings: n-max n-max n-large n-large n-medium n-medium n-small n-small zero zero p-small p-small p-medium p-medium p-large p-large p-max p-max Constraint: sub (dt 1 V) (dt 0 qi) (dt 0 qo) NumVarsToPrint: 3 VarsToPrint: V qi qo

  16. A JMorven Quantity Space NumQSpaces: 2 QSpaceName: tanks-quantity-space NumQuantities: 9 n-max -1 -1 0 0.1 n-large -0.9 -0.75 0.05 0.15 n-medium -0.6 -0.4 0.1 0.1 n-small -0.25 -0.15 0.1 0.15 zero 0 0 0 0 p-small 0.15 0.25 0.15 0.1 p-medium 0.4 0.6 0.1 0.1 p-large 0.75 0.9 0.15 0.05 p-max 1 1 0.1 0 QSpaceName: tanks-quantity-space2 NumQuantities: 5 nl-dash -1 -0.75 0 0.15 ns-dash -0.6 -0.15 0.1 0.15 zero 0 0 0 0 ps-dash 0.15 0.6 0.15 0.1 pl-dash 0.75 1 0.15 0

  17. Possible States state vector state vector 1 + + + + 22 + - o + 2 + + + o 23 + - o o 3 + + + - 24 + - o - 4 + + o + 25 + - - + 5 + + o o 26 + - - o 6 + + o - 27 + - - - 7 + + - + 28 o + + + 8 + + - o 29 o + + o 9 + + - - 30 o + + - 10 + o + + 31 o + o + 11 + o + o 32 o + o o 12 + o + - 33 o + o - 13 + o o + 34 o + - + 14 + o o o 35 o + - o 15 + o o - 36 o + - - 16 + o - + 37 o o + + 17 + o - o 38 o o + o 18 + o - - 39 o o + - 19 + - + + 40 o o o + 20 + - + o 41 o o o o 21 + - + -

  18. Step Response V t

  19. qi 14 7 21 30 V Solution Space

  20. Soundness and Completeness • Sound • Guarantees to find all possible behaviours of system • Incomplete • Unfortunately also finds non-existent (spurious) behaviours • Still useful for ascertaining that a dangerous state cannot be reached. • Large research effort to remove spurious behaviours • we will skim the surfarce of the surface!

  21. qi V qo qi t Single Tank System: Ramp Input plane 0 qO = kV V’ = qi - qO plane 1 q’O = kV’ V’’ = q’i - q’O plane 2 q’’O = kV’’ V’’’ = q’’i - q’’O • Input: Stepped Ramp

  22. 32 21 12 3 7 5 30 34 2 Element Vector Envisionment

  23. 21 12 5 3 7 30 32 34 3 Element Vector Envisionment

  24. Distinct Behaviours V 21 12 5 3 7 3 34 t 32 30

  25. 7 5 34 12 3 32 21 30 Solution Space qi V

  26. Total Solution Space: Single Compartment

  27. Tank A Tank B Cascaded Systems qi plane 0 qx = k1.h1 qo = k2.h2 h1’ = qi - qx h2’ = qx - qo plane 1 qx’ = k1.h1’ qo’ = k2.h2’ h1’’ = qi’ - qx’ h2’’ = qx’ - qo’ plane 2 qx’’ = k1.h1’’ qo’’ = k2.h2’’ h1’’’ = qi’’ - qx’’ h2’’’ = qx’’ - qo’’ h1 qx qo h2 u 1 2 k12.x1 k20.x2

  28. Cascaded Systems Envisionment 11 7 3 1 4 12 8 0 9 5 10 13 2 6

  29. Cascaded Systems Solution Space h1’=0 h2 h1’=0 3 7 11 1 4 12 8 5 9 13 h1 0 2 6 10

  30. Complete Solution Space: Cascaded Compartments

  31. Categorisation of Behaviours Behaviours Real Spurious Actual Potential Chattering Non-chattering

  32. Fuzzy Set Theory and FQR • Two main concepts: the cut and the Approximation principle • The cut A = [p1, p2, p3, p4] Aa= [p1+p3(-1), p2+p4(1-)]

  33. Representational Primitives

  34. Representational Primitives (2) • Functional primitives • More specific than M+/- relations, though still incomplete • Compiled (tabular) set of fuzzy if-then rules - permits incusion of empirical information • Derivative primitive

  35. The Approximation Principle The Approximation principle facilitates the mapping of the result of a fuzzy operation onto the values in the quantity space of the result variable. A measure of the Goodness of Approximation is achieved by means of a Distance Metric d(A, A’) = [(power(A)-power(A’))2+(centre(A)-(centre(A’))2]0.5 power([a,b,a,b]) = 0.5[2(a+b) + a + b)] centre([a,b,a,b]) = 0.5[a+b]

  36. Approximation Principle (2)

  37. Transition Rules

  38. Temporal Calculations

  39. Fuzzy Vector Envisionment

  40. Experimental Test

  41. Fuzzy Vector State Labels

  42. FVE Graph for a Step Input

  43. Fuzzy Qualitative Behaviours

  44. h2 h2 Huge Large Large Medium Small Small Small Large h1 Small Medium Large Huge h1 Cascaded System

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