Foundations of Modelling and Simulation Hans Vangheluwe Modelling, - PowerPoint PPT Presentation

Foundations of Modelling and Simulation Hans Vangheluwe Modelling, Simulation and Design Lab (MSDL) School of Computer Science, McGill University, Montr´ eal, Canada Department of Mathematics and Computer Science, University of Antwerp, Belgium Hans Vangheluwe Modelling and Simulation Foundations 1

Language Engineering and Simulation Hans Vangheluwe Modelling and Simulation Foundations 2

Hierarchy of System Specification of Structure and Behaviour • Basis of System Specification: sets theory, time base, segments and trajectories • Hierarchy of System Specification ( causal , deterministic ) 1. I/O Observation Frame 2. I/O Observation Relation 3. I/O Function Observation 4. I/O System • Multicomponent Specifications • Non-causal models ref: Waymore, Zeigler, Klir, . . . Hans Vangheluwe Modelling and Simulation Foundations 3

Set Theory Properties: { 1 , 2 , . . . , 9 } { a, b, . . . , z } N , N + , N + ∞ R , R + , R + ∞ EV = { ARRIV AL, DEPARTURE } EV φ = EV ∪ { φ } Structuring: A × B = { ( a, b ) | a ∈ A, b ∈ B } G = ( E, V ) , V ⊆ E × E Hans Vangheluwe Modelling and Simulation Foundations 4

Comparing things Hans Vangheluwe Modelling and Simulation Foundations 5

Nominal Scale: e.g., gender A scale that assigns a category label to an individual. Establishes no explicit ordering on the category labels. Only a notion of equivalence “ = ” is defined with properties: 1. Reflexivity: x = x ∨ x � = x . 2. Symmetry of equivalence: x = y ⇔ y = x . 3. Transitivity: x = y ∧ y = z → x = z . Hans Vangheluwe Modelling and Simulation Foundations 6

Ordinal Scale: e.g., degree of happiness A scale in which data can be ranked , but in which no arithmetic transformations are meaningful. It is meaningless to talk about difference (distance). In addition to equivalence, a notion of order < is defined with properties: 1. Symmetry of equivalence: x = y ⇔ y = x . 2. Asymmetry of order: x < y → y � < x . 3. Irreflexivity: x � < x . 4. Transitivity: x < y ∧ y < z → x < z . Hans Vangheluwe Modelling and Simulation Foundations 7

Partial ordering The ordering may be partial (some data items cannot be compared). t3 t2 t1 t6 t7 t4 t5 The ordering may be total (all data items can be compared). ∀ x, y ∈ X : x < y ∨ y < x ∨ x = y Hans Vangheluwe Modelling and Simulation Foundations 8

Interval Scale: e.g., Shoe Size A scale where distances between data are meaningful. On interval measurement scales, one unit on the scale represents the same magnitude of the characteristic being measured across the whole range of the scale. Interval scales do not have a “true” zero point, however, and therefore it is not possible to make statements about how many times higher one value is than another. In addition to equivalence and order, a notion of interval is defined. The choice of a zero point is arbitrary. Hans Vangheluwe Modelling and Simulation Foundations 9

Ratio Scale: e.g., age Both intervals between values and ratios of values are meaningful. A meaningful zero point is known. “A is twice as old as B”. Hans Vangheluwe Modelling and Simulation Foundations 10

Time Base • Simulation of Dynamic Systems: irreversible passage of time . • Time Base T : – { NOW } (instantaneous) – R : continuous-time – N or isomorphic: discrete-time • Ordering: – Ordinal Scale (possibly partial ordering) – Interval Scale – Ratio Scale Hans Vangheluwe Modelling and Simulation Foundations 11

Time Bases for hybrid system models Hans Vangheluwe Modelling and Simulation Foundations 12

Time Bases for hybrid system models TD (tc, td) TC “nested time” for nested experiments. Hans Vangheluwe Modelling and Simulation Foundations 13

Behaviour ≡ Evolution over Time • With time base, describe evolution over time • Time function, trajectory , signal: f : T → V • Restriction to T ′ ⊆ T f | T ′ : T ′ → V , ∀ t ∈ T ′ : f | T ′ ( t ) = f ( t ) – Past of f : f | T t � – Future of f : f | T � t • Restriction to an interval: segment ω : � t 1 , t 2 � → V Hans Vangheluwe Modelling and Simulation Foundations 14

Types of Segments V continuous T V piecewise continuous T V piecewise constant T V discrete event T Hans Vangheluwe Modelling and Simulation Foundations 15

Cashier-Queue System Departure Arrival Cashier Queue Physical View Departure Arrival Cashier Queue [IAT distribution] [ST distribution] Abstract View Hans Vangheluwe Modelling and Simulation Foundations 16

Trajectories Input Events Arrival 10 20 30 40 50 T E1 E2 queue_length 2 1 0 10 20 30 40 50 T state= queue_length x cashier_state cashier_state Busy Idle 10 20 30 40 50 T Output Events Departure 10 20 30 40 50 T E3 E4 Hans Vangheluwe Modelling and Simulation Foundations 17

I/O Observation Frame (causal) O = � T, X, Y � • T is time-base : N (discrete-time), R (continuous-time) • X input value set: R n , EV φ • Y output value set: system response Hans Vangheluwe Modelling and Simulation Foundations 18

I/O Relation Observation IORO = � T, X, Ω , Y, R � • � T, X, Y � is Observation Frame • Ω is the set of all possible input segments • R is the I/O relation Ω ⊆ ( X, T ) , R ⊆ Ω × ( Y, T ) ( ω, ρ ) ∈ R ⇒ dom ( ω ) = dom ( ρ ) • ω : � t i , t f � → X : input segment • ρ : � t i , t f � → Y : output segment • note: not really necessary to observe over same time domain Hans Vangheluwe Modelling and Simulation Foundations 19

I/O Function Observation IOFO = � T, X, Ω , Y, F � • � T, X, Ω , Y, R � is a Relation Observation • Ω is the set of all possible input segments • F is the set of I/O functions f ∈ F ⇒ f ⊂ Ω × ( Y, T ) , where f is a function such that dom ( f ( ω )) = dom ( ω ) • f = initial state : unique response to ω • R = � f ∈ F f Hans Vangheluwe Modelling and Simulation Foundations 20

I/O System • From Descriptive Variables (properties) to State . • State summarizes the past behaviour of the system. • Future is uniquely determined by – current state – future input Hans Vangheluwe Modelling and Simulation Foundations 21

SY S = � T, X, Ω , Q, δ, Y, λ � T time base input set X ω : T → X input segment state set Q δ : Ω × Q → Q transition function output set Y λ : Q → Y (or Q × X → Y ) output function ∀ t x ∈ [ t i , t f ] : δ ( ω [ t i ,t f ] , q i ) = δ ( ω [ t x ,t f ] , δ ( ω [ t i ,t x ] , q i )) Hans Vangheluwe Modelling and Simulation Foundations 22

Composition Property Q δ (t_x -> t_f) δ (t_i -> t_x) δ (t_i -> t_f) T X ω [t_x, t_f] ω [t_i, t_x] ω [t_i, t_f] t_i t_x t_f T Hans Vangheluwe Modelling and Simulation Foundations 23

Simulator: step through time Y λ λ δ Q δ δ ω ω ω X T ti tf Hans Vangheluwe Modelling and Simulation Foundations 24

Formalism classification based on general system model T: Continuous T: Discrete T: { NOW } Q : Continuous ODE, DEVS Difference Eqns. (DTSS) Algebraic Eqns. Q : Discrete Discrete-event Finite State Automata Integer Eqns. Basis for general, standard software architecture of simulators Further classifications based on structure of formalisms (in particular of δ ) Hans Vangheluwe Modelling and Simulation Foundations 25

Rule-based specification of δ Rule 1 (priority 3) Locate Initial Current State 1 1 ::= <ANY> <COPIED> 3 2 Current State Rule 2 (priority 1) State Transition 1 1 Current State Current State condition: 2 2 matched(4).input == input[0] 4 3 <COPIED> <COPIED> / ::= 3 5 5 4 <COPIED> <ANY> / <ANY> <ANY> <ANY> <COPIED> action: remove(input[0]) Rule 3 (priority 2) Local State Transition 1 1 Current State Current State condition: matched(4).input == input[0] 2 2 ::= 4 4 <COPIED> / <COPIED> <ANY> / <ANY> 3 3 action: <COPIED> <ANY> remove(input[0]) Hans Vangheluwe Modelling and Simulation Foundations 26

System under study: T, h controlled liquid closed fill empty is_full is_empty is_cold heat is_hot off cool Hans Vangheluwe Modelling and Simulation Foundations 27

Detailed (continuous) view, ALG + ODE Inputs (discontinuous → hybrid model): • Emptying, filling flow rate φ • Rate of adding/removing heat W  Parameters: 1 dT l [ W = cρA − φ ( T − T in )]  dt  • Temperature of influent T in   dl  = φ   dt • Cross-section surface of vessel A    = ( l < l low )  is low  • Specific heat of liquid c = ( l > l high ) is high • Density of liquid ρ     State variables:  = ( T < T cold ) is cold     • Temperature T  = ( T > T hot )  is hot  • Level of liquid l Outputs (sensors): • is low, is high, is cold, is hot Hans Vangheluwe Modelling and Simulation Foundations 28