1. CS 367: Model-Based Reasoning Lecture 2 (01/15/2002)
Gautam Biswas

2. Today’s 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(?))

3. Lecture 1: (Home Work)
Additional reading material: F.E. Cellier, H. Elmqvist, and M. Otter, “Modeling from Physical Principles,” (pdf file URL:
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?

4. 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

5. 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

6. 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.
Dynamic System, S
State Variables, X
Output Variables Y
Input Variables U

7. Example: Energy-based Modeling of Systems

8. Deriving the ODE model

9. Example: Discrete-State Modeling of Systems
Warehouse Systems
Arriving Products
u1(t)
Departing Products
u2(t)
x(t)
x(t) :arrival; u1(t) = 1; u2(t) = 0
x(t+) = x(t) :departure; u1(t) = 0; u2(t) = 1; x(t) > 0
x(t) :otherwise
Question: Is this similar to a tank system?
What is the difference?

10. 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

11. 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
D/A
energy domain
control
algorithm
plant
actuators
sensors
physical system
embedded controller + -
Tsample
A/D

12. Supervisory Controller
Decision
Maker
Environment
Switching
Signal u1
Controller 1 u2
Controller 2 u y
Plant • um
Controller m

13. Example: Bouncing Ball
ball position – x1 ball velocity – x2 acceleration – g
coefficient of restitution – c  [0,1]
x1 > 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

14. 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.

15. 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:
Continuous time systems versus Discrete time systems

16. Levels of Abstraction in a Discrete Event System

17. 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

18. 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

19. Major System Classifications
Time-Driven
Continuous-State
Event-Driven
Systems
Dynamic
Linear
Time-varying
Static
Time-invariant
Nonlinear
Discrete-State
stationary
DES
sampled
Stochastic
Deterministic
Discrete-Time
Continuous-Time

20. 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

21. 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

22. 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

23. Automaton Model for two philosophers
Notion of parallel composition

24. Recursive Equation Model: Two philosopher problem