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Embedded Systems ECEC 356 – Fall 2014

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## Embedded Systems ECEC 356 – Fall 2014

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**Embedded SystemsECEC 356 – Fall 2014**• Chapter 4: Extended State Machines and Hybrid Automata • Chapter 5: Composition of machines Andrew R. Cohen 10/14/2014**Last time**• - what’s on the test? You tell me… • Today • - review (1-4, etc) • - start chapter 5 • - hw3 due • - Next time • - exam 1**Example: Mouse Double Click Detector**This simple form of hybrid system is called a timed automaton, where the dynamics is just passage of time.**Example: Mouse Double Click Detector**• How many states does this automaton have?**Timed automaton model of a traffic light controller**This light remains green at least 60 seconds, and then turns yellow if a pedestrian has requested a crossing. It then remains red for 60 seconds.**Example: “Tick” Generator (Timer)**• How would you model a timer that generates a ‘tick’ • each time T time units elapses? • A similar timed automaton can model a generator • of a timer interrupt.**Hybrid Automaton for Bouncing Ball**• y – vertical distance from ground (position) • a – coefficient of restitution, 0 ·a· 1 • If you plotted y(t), what would it look like?**Hybrid Automaton for Bouncing Ball**• y – vertical distance from ground (position) • a – coefficient of restitution, 0 ·a· 1**When do reactions occur in a hybrid automaton?**• Reactions are occurring continually, with the continuous state variable x being continually updated.**When do reactions occur in a hybrid automaton?**• Suppose x and y are discrete and pure signals. When does the transition occur? • Answer: at the earliest time t when x is absent after entering s1.This will always be the same time when s1 is entered. Why? • If x is absent when s1 is entered, then the transition is taken then.If x is present when s1 is entered, then it will be absent at a timeinfinitesimally larger. How to model this rigorously?**Example: Newton’s Cradle**• Image src: Wikipedia Commons • A middle ball does not move, so its momentum must be 0. • But the momentum of the first ball is transferred somehow to the fifth. So there is an instant at which it is non-zero!**Sneak Preview: Super-Dense Time**• We will solve this problem by modeling signals as functions of super-dense time: • In this way, x can be present and absent at the same time, with a well-defined order between its presence and absence.**Modeling with State Machines**• Questions: • How to represent the system for: • Systematic analysis • So that a computer program can manipulate it • How to model its environment • How to compose subsystems to make bigger systems • How to check whether the system satisfies its specification in its operating environment**Actor Model for State Machines**• Expose inputs and outputs, enabling composition:**Composition of State Machines**• Side-by-side composition • Cascade composition • Feedback composition**Side-by-Side Composition**A key question: When do these machines react? Two possibilities: • Together (synchronous composition) • Independently (asynchronous composition)**Note that these two states are not reachable.**Synchronous Composition Synchronous composition**Asynchronous composition**using interleaving semantics Asynchronous Composition Note that now all states are reachable.**Syntax vs. Semantics**Synchronous or Asynchronous composition? If asynchronous, does it allow simultaneous transitions in A & B?**Asynchronous composition is not synchronous composition with**stuttering transitions. Stuttering is a reaction where no output is produced and the state is not changed. Synchronous composition These two FSMs cannot stutter. If they react, they change state.**Asynchronous composition is not synchronous composition with**stuttering transitions. Stuttering is a reaction where no output is produced and the state is not changed. Asynchronous composition with interleaving semantics These two FSMs cannot stutter. If they react, they change state.**Asynchronous Composition**• M1 = (S1, I1, O1, U1, s10) and M2 = (S2, I2, O2, U2, s20) • M is the asynchronous composition of M1 and M2 • = (S1× S2, I1× I2, O1× O2, U, (s10, s20)) • where • U((s1, s2), (i1, i2)) = ((s1’, s2’), (o1, o2)) • and • (s1’, o1) = U1(s1, i1) AND s2’ = s2 & o2 is absent • OR (s2’, o2) = U2(s2, i2) AND s1’ = s1 & o1 is absent (note interleaving semantics)**Cascade Composition**A B Output port(s) of A connected to input port(s) of B**Example: Pedestrian Light**This light stays green for 55 seconds, then goes red. Upon receiving a sigR input, it repeats the cycle.**Pedestrian Light with Car Light**sigY sigG sigR sigR pedG What is the size of the state space of the composite machine? pedR**Homework**• Exam 1 – Thursday 10/16/2014 • Chapters 1-4 of L&S, Appendix A • All material from lab • Open book / notes – NO ELECTRONIC DEVICES • Next Week • Chapter 5 – composition of state machines**Instructor Contact Information**Andrew R. Cohen Associate Prof. Department of Electrical and Computer Engineering Drexel University 3120 – 40 Market St., Suite 110 Philadelphia, PA 19104 office phone: (215) 571 – 4358 http://bioimage.coe.drexel.edu/courses acohen@coe.drexel.edu