CS 367: ModelBased Reasoning Lecture 2 01152002

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CS 367 Model-Based ReasoningLecture 2
(01/15/2002)

• Gautam Biswas

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Todays Lecture

• Introductory Lecture (01/10) Modeling of Systems
  and its applications
• Examples of Discrete Event, Continuous, and
  Hybrid Systems
• Topic 1(2-3 weeks) Discrete Event Modeling of
  Systems (ref S. Lafortune, et al. Automata
  based models, Petri Net based models(?))

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Lecture 1 (Home Work)

• Additional reading material F.E. Cellier, H.
  Elmqvist, and M. Otter, Modeling from Physical
  Principles, (pdf file URL http//www.vuse.vander
  bilt.edu/biswas/Courses/cs367/papers)
• Problems
• 1. The height of water in a reservoir fluctuates
  with time. If you had to construct a dynamic
  system model to help water resource planners
  predict variations in the height, what input
  quantities would you consider? How many state
  variables would you need in your model?
• 2. Suppose you were a heating engineer and you
  wished to consider your house as a dynamic
  system. Without a heater the average temperature
  of the house would clearly vary over a 24 hour
  period. What might you consider as state
  variables for a simple dynamic model? How would
  you expand your model to predict the temperatures
  in several rooms in your house? How does the
  installation of a thermostat controlled heater
  change your model?

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Systems Viewpoint to Modeling

• Model to study operation of complete system as
  opposed to operation of the individual parts
• Method of model building compositionality
• Method of analysis isolate into parts
• Unified approach to modeling, rather than being
  domain-specific
• Energy-based modeling of physical systems
• Discrete-event models of systems change in
  system directly linked to the occurrence of
  events
• Combine modeling paradigms Hybrid Systems
  approach to modeling

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Modeling of Dynamic Systems

• State-Determined Systems -- our goal is to start
  with physical component descriptions of systems
  understanding of component behavior to create
  mathematical models of the system.
• Mathematical model of state-determined system
  defined by set of ordinary differential equations
  on the so-called state variables. Algebraic
  relations define values of other system variables
  to state variables.
• Dynamic behavior of state-determined system
  defined by (i) values of state variables at some
  initial time, and (ii) future time history of
  input quantities to system.
• In other words, our system models satisfy
  the Markov property

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Uses of Dynamic Models

• Analysis for prediction, explanation,
  understanding, and control. Two types (a)
  analytic methods, and (ii) simulation-based
  methods. Given S, X at present, and U for the
  future, predict future X and Y.
• Identification. Given U and Y find S and X
  consistent with U and Y. (under normal and faulty
  conditions)
• Synthesis. Given U and a desired Y, find S such
  that S acting on U produces Y.

Output Variables Y
Dynamic System, S State Variables, X
Input Variables U
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Example Energy-based Modeling of Systems
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Deriving the ODE model
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Example Discrete-State Modeling of Systems
Warehouse Systems
Question Is this similar to a tank system?
What is the difference?
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What is A Hybrid System?

• Dynamic systems that require more than one
  modeling language to characterize their dynamics
• Provide a mathematical framework for analyzing
  systems with interacting discrete and continuous
  dynamics
• Capture the coupling between digital computations
  and analog physical plant and environment
• Continuous Dynamics mechanical, fluid, thermal
  systems, linear circuits, chemical reactions
• Discrete dynamics collisions, switches in
  circuits, valves and pumps

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Hybrid Models of Physical Systems

• Why Hybrid Models?
• Proliferation of Embedded Systems
• Simplify Behavior analysis of complex non linear
  systems
• Motivation(s)
• Monitoring Diagnosis
• Design, Control

signal domain
energy domain
D/A
A/D
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Supervisory Controller
Environment
Decision Maker
Switching Signal
u1
Controller 1
u2
Controller 2
y
u
Plant

um
Controller m
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Example Bouncing Ball
ball position x1 ball velocity x2
acceleration g coefficient of restitution c ?
0,1 x1 gt 0 ? continuous flow governed by
differential equation when transition condition
satisfied ? discrete jump occurs Behavior is
zeno, i.e., infinite number of bounces occur in
finite time interval
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Example Thermostat
Thermostat (controller) turns on radiator between
68 70 degrees and turns off the radiator
between 80 and 82 degrees. Result non
deterministic system for a given initial
condition there are a whole family of different
executions.
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Discrete Event Systems (Chapter 2 Cassandras
and Lafortune Languages and Automata)

• What is a discrete event system?
• State space of system discrete
• State transitions are only observed at discrete
  points in time, i.e., state transitions are
  associated with events
• For multimedia overview of discrete event systems
  look up http//vita.bu.edu//cgc/MIDEDS
• Continuous time systems versus Discrete time
  systems

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Levels of Abstraction in a Discrete Event System
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Informal Definition of Event

• Specific action
• e.g., turn switch on/off
• Spontaneous occurrence dictated by nature of
  environment
• e.g., power supply goes off
• Certain conditions being met within system

height of liquid in tank, h ? h0 ? flow through
pipe
h0
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Time-driven versus Event-Driven Systems

• Time-driven Synchronous at every clock tick
  event occurs which advances system behavior
• Event-driven Asynchronous or concurrent at
  various time instances not necessarily known in
  advance and not necessarily coinciding with clock
  ticks event e announces its occurrence

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Major System Classifications
stationary
DES
sampled
Stochastic
Deterministic
Discrete-Time
Continuous-Time
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State Evolution in DES

• Sequence of states visited
• Associated events cause the state transitions
• Formal ways for describing DES behavior, i.e.,
  what is a language for describing DES behavior?
• Automata
• Petri Nets

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Languages for DES behavior

• Simplest a timed language where timing
  information has been deleted
• untimed modeling formalism defined by event
  sequence e1 e2 en
• Timed Language set of all timed sequences of
  events that the DES can generate/execute
• (e1,t1) (e2,t2) (en,tn)
• Stochastic Timed Language a timed langauge with
  a probability distribution function defined over
  it

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Discrete Event Modeling Formalisms

• State-based define a state space and specify a
  state-transition structure
• (out_state, event, in_state) triples
• e.g., Automata and Petri Nets
• Trace-based based on (recursive) algebraic
  expressions
• e.g., Communicating Sequential Processes (CSPs)
• We will study modeling, analysis, and supervisory
  control with untimed and timed automata

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Automaton Model for two philosophers
Notion of parallel composition
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Recursive Equation Model Two philosopher problem