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George M. Coghill The Morven Framework

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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

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

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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

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

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The Morven Framework Constructive Non-constructive Simulation Envisionment Synchronous Asynchronous

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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

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4-tuple fuzzy numbers (a, b, ) precise and approximate useful for computation x A (x) 1 0 a x (a) A (x) 1 0 a b x (b) A (x) 1 0 a- a x a+ (c) A (x) 1 0 a- b+ ab (d) A convenient fuzzy representation

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Fuzzy Quantity Spaces A (x) 10 x 0.20.4 0.6 0.8-0.8-0.6 -0.4 -0.2 0

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Curve Shapes + 0 0 + _ _ d1d1 d2d2

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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. [++][+o][+-] [o+][oo][o-] [-+][-o][- -]

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Single Tank System V qiqi qoqo plane 0 q O = kV V = q i - q O plane 1 q O = kV V = q i - q O plane 2 q O = kV V = q i - q O

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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 1 u k 10.x 1

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Models in Morven (define-fuzzy-model (short-name ) (variables ) (auxiliary-variables ) (input ) (constraints (print ) )

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

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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 -0.150.1 0.15 zero 0 0 00 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 -0.750 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 0.751 0.15 0

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Possible States statevectorstatevector 1+ + + +22+ - o + 2+ + + o23+ - o o 3+ + + -24+ - o - 4+ + o +25+ - - + 5+ + o o26+ - - o 6+ + o -27+ - - - 7+ + - +28o + + + 8+ + - o29o + + o 9+ + - -30o + + - 10+ o + +31o + o + 11+ o + o32o + o o 12+ o + -33o + o - 13+ o o +34o + - + 14+ o o o35o + - o 15+ o o -36o + - - 16+ o - +37o o + + 17+ o - o38o o + o 18+ o - -39o o + - 19+ - + +40o o o + 20+ - + o41o o o o 21+ - + -

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Step Response t V

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Solution Space 21 14 7 30 V qiqi

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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!

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Single Tank System: Ramp Input V qiqi qoqo t qiqi Input: Stepped Ramp plane 0 q O = kV V = q i - q O plane 1 q O = kV V = q i - q O plane 2 q O = kV V = q i - q O

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

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3 Element Vector Envisionment 2112 3 30 5 32 7 34

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

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Solution Space 21 730 V qiqi 12 35 3234

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Total Solution Space: Single Compartment

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Cascaded Systems 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 Tank A Tank B 1 2 u k12.x1 k20.x2 h1h1 h2h2 qiqi qxqx qoqo

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

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Cascaded Systems Solution Space h2h2 h1h1 h 1 =0 1 11 12 6 2 0 10 13 9 8 7 5 3 4

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Complete Solution Space: Cascaded Compartments

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

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Fuzzy Set Theory and FQR Two main concepts: the cut and the Approximation principle The cut A = [p1, p2, p3, p4] A = [p1+p3( p2+p4(1-

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Representational Primitives

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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

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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, = 0.5[2(a+b) + centre([a,b, = 0.5[a+b]

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Approximation Principle (2)

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Transition Rules

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Temporal Calculations

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Fuzzy Vector Envisionment

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Experimental Test

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Fuzzy Vector State Labels

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FVE Graph for a Step Input

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Fuzzy Qualitative Behaviours

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Cascaded System Small Large h2h2 h1h1 Small Medium Large Huge h2h2 h1h1

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