1. CSCI1600: Embedded and Real Time Software
Lecture 9: Modeling III: Hybrid Systems
Steven Reiss, Fall 2016
Talk about Xeno’s paradox when describing bouncing ball.
Add other type of timed automata?

2. Time and Finite State Automata
We’ve added timing to our automata in an ad-hoc fashion
Assuming a fixed period
Added timer events
Indicated length of time in a state (min/max)
No notion of absolute time, clocks, etc.
These are easy to implement
In order to prove properties of the system
With respect to time
We need a cleaner, more formal representation
Assuming time is continuous
Lecture 9: Modeling Hybrid Systems
2/22/2019

3. Timed Automata Basic idea Timer variables can be set explicitly
Extended finite state machine
Add the notion of timer variables
Timer variables can be set explicitly
All timer variables are periodically incremented
Can think of this as every k milliseconds
Can think of this as continuous as well
Timer variables can be used in conditions
No ad-hoc time constraints
Lecture 9: Modeling Hybrid Systems
2/22/2019

4. Timed Automata Assume X is a timer variable. What does this do?
Lecture 9: Modeling Hybrid Systems
2/22/2019

5. Another example
Lecture 9: Modeling Hybrid Systems
2/22/2019

6. Extended timed automata
Lecture 9: Modeling Hybrid Systems
2/22/2019

7. Timed Automata Are these still finite state?
Timer variables can be unbounded
Timer variables are continuous
Determine the state ranges for times
These form time-states
Can create an equivalent FSA
Lecture 9: Modeling Hybrid Systems
2/22/2019

8. Modeling and the Real World
Often we want to model both the system and the world
The system is a finite automata
The real world is continuous & not just clocks
Problems
How to combine the two
What can we do with the combination
Understanding
Proving properties
Lecture 9: Modeling Hybrid Systems
2/22/2019

9. Modeling in the Real World
Lecture 9: Modeling Hybrid Systems
2/22/2019

10. Hybrid Automata State variables reflect the real world
Updates done over time automatically
Updates can use continuous modeling
Can specify velocity, acceleration rather than position
Can specify rules for doing the update
Lecture 9: Modeling Hybrid Systems
2/22/2019

11. Bouncing Ball Example Will the ball stop bouncing (Zeno’s paradox)
What time do we use?
Lecture 9: Modeling Hybrid Systems
2/22/2019

12. RC Car
Lecture 9: Modeling Hybrid Systems
2/22/2019

13. Hybrid Timed Automata
The external (timed) portion is set outside the automata
Rules for continuous update
Model the physics of the real world
Lecture 9: Modeling Hybrid Systems
2/22/2019

14. Why Model the World?
Does the code need to know the state of the real world
Either precisely or approximately
Does it need to compute that state
Or retrieve it via sensors
How accurate can you model the world?
Consider speed control on a car
Proving properties of the program wrt the model
Does this prove the system correct
Lecture 9: Modeling Hybrid Systems
2/22/2019

15. Implementation Approach
Get information from the real world
What the actual speed is, change of speed
Actual temperature (and change rate)
Compute the difference between actual and desired
This is called the error
Use this to determine the action
This is considered feedback
Lecture 9: Modeling Hybrid Systems
2/22/2019

16. Control Theory Why are computers called “digital computers”
What is the alternative
What is an analog computer
It turns out that analog computers are easy to build
Either mechanically or electronically
Predate digital computers
Lecture 9: Modeling Hybrid Systems
2/22/2019

17. Analog Computers
Lecture 9: Modeling Hybrid Systems
2/22/2019

18. Analog Computer Can do simple math operations
Some complex math operations are easy
Integration, differentiation
Lecture 9: Modeling Hybrid Systems
2/22/2019

19. Controlling Physical Systems
Analog computers are designed for this
If you want a car to go in a straight line
Use feedback from left/right to update steering
Use intergrator to determine error to control system
Analog control is all about feedback
We can do the same in digital systems
Covered a week from Friday
First we need to cover details for the assignment
And some additional modeling concepts
Lecture 9: Modeling Hybrid Systems
2/22/2019

20. Homework
Read Chapter 7.1
Lecture 9: Modeling Hybrid Systems
2/22/2019