Leveraging Synchronous Language Principles for Hybrid System Models

1
Leveraging Synchronous Language Principles for
Hybrid System Models

• Haiyang Zheng and Edward A. Lee
• UC Berkeley

2
Introduction

• A lot of tools have been developed for supporting
  model-based designs of hybrid systems.
• Charon
• Dymola/Modelica
• Simulink
• HyVisual/Ptolemy
• All these tools use a block-diagram syntax to
  represent models.

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Main-Stream Operational Semantics

• The main-stream operational semantics for
  simulating DE, CT, and hybrid systems models
  heavily rely on the execution order of blocks.
• The execution order is usually determined by
  topological sorting based on data dependencies.
• It is not unique.
• The execution order affects simulation results.
• Topological sorting is not always possible.

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Operational Semantics Discrete Event Example

• DE operational semantics
• A block executes whenever receiving a trigger
  event in any input port.
• When executing, if an input port has no present
  event, the block uses an absence event for
  computation.

We get either 1 followed by 2, or 3 depending on
the execution order of Scale and AddSubtract
blocks.
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Operational Semantics Discrete Event Example
(continued)

• The execution order may cause unexpected delay
  among simultaneous events.
• When this delay interferes with user specified
  time delay and feedback loops, the simulation
  results are more complicated.

I leave this an exercise for readers to figure
out how many possible outputs there are.
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Operational Semantics Continuous Time Example

• CT operational semantics
• Determined by the actual ODE solvers used for
  solving dynamic equations
• Example Runge-Kutta 2-3 Solver
• Given x(tn) and a step size h to solve x(tn1),
  calculate
• then let

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Continuous Time ExampleHow does Integration
work?
It takes three micro steps for the RK 2-3 solver
to finish a complete integration. The outputs
generated at a micro step by one integrator
affect the inputs of other integrators.
x
dx/dt
y
dy/dt
x
At each micro step, perform the followings in
order
dx/dt
Update dx/dt
Update dy/dt
y
Update x
Update y
dy/dt
t
t ?
t ?/2
t 3?/4
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So,

• Topological sorting is important for correctness
  of simulation.
• But getting topological sorting correct is hard.

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Hierarchical Execution Continuous Time Example
A designer expects certain invariants
transformations of a model by adding hierarchies
do not change behavior.
A correct result
An incorrect result in HyVisual releases earlier
than version 5.0.
Results are calculated with the RK 2-3 solver.
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Topological Sorting of Composites

• Introducing hierarchies makes topological sorting
  hard.

How to sort the composite actor without
flattening the hierarchy?
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Causality Interface A Helper for Topological
Sorting

• Causality interfaces expose dependency
  information.

Only port4 depends on port1.
More information can be found in FIT2005, Lee,
Zheng, Zhou
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Synchronous Languages

• The model of computation is called synchronous
  reactive (SR).
• Esterel
• Lustre
• SCADE (visual editor for Lustre)
• Signal
• Statecharts (some variants)
• Ptolemy II SR domain
• A strong formal property Execution results do
  NOT depend on schedule or topological sorting!

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SR Operational SemanticsMathematics Foundation

• Requirements
• All possible signal values form a flat CPO with
  ?, called unknown, as the bottom element.
• All blocks are monotonic functions (and therefore
  continuous functions and composable).
• Fixed Point Theorem Introduction to lattices and
  order, Davey and Priestley, 2002
• Let f A ? A be a continuous function defined on
  a CPO A, then f has a least fixed point given by
  ?n?0 f n (?).
• The number n is finite.

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SR Operational SemanticsExecution Algorithm

• At a tick t, reset all signal values to unknown,
  ?.
• Pick up a block (no preference needed) and
  evaluate its output signal values based on its
  current input signal values and states.
• Repeat step 2 until either all signal values have
  been resolved or no more signal values can be
  resolved.
• The set of all resolved signal values is called
  the least fixed point of the model at tick t.
• Advance to another tick t t1, repeat step 1.

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SR Operational Semantics A Simple Example
? 0
? 0
? absent
?

• Invoke AddSubtract
• Invoke Ramp
• Invoke NonStrictDelay
• Invoke AddSubtract
• Invoke NonStrictDisplay
• Invoke NonStrictDisplay2
• Fixed point is reached with all signal values
  resolved.

?
One possible execution schedule
?
?
?
?
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So,

• Topological sorting is not that important for
  correctness of simulation but a very useful
  optimization.

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An Operational Semantics For Hybrid Systems

• Leverage the SR Operational Semantics to develop
  a general operational semantics for DE, CT, FSM,
  SR, and Hybrid System models
• Separate the concern of simulation correctness
  from the other considerations for implementations
  details, in particular, the topological sorting.
• Benefits
• The correctness of individual implementations can
  be verified.
• Treat topological sorting as an optimization,
  which helps to improve the execution performance.

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What We Do

• Augment the value set with unknown, ?.
• Ensure an actor implements a monotonic function
• Explicitly specify strictness and non-strictness
  property
• Write actors according to the abstract actor
  semantics
• Associate time semantics to intervals between
  ticks.

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Abstract Actor Semantics

• iterate()

• prefire()
• fire()
• postfire()

• Flow of control
• Initialization
• Execution
• Finalization
• Specifications
• prefire() synchronizes to the environment and
  checks firing conditions (such as strictness)
• fire() generates outputs based on current inputs
  and states
• postfire() updates the states for next iteration

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Time Semantics Between Ticks

• We use super-dense time, for hybrid
  systems.
• The interval between two ticks represents an
  increment of index or real-time.

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Operational SemanticsExecution Algorithm

• At a time t, reset all signal values to unknown,
  ?.
• Pick up a block (no preference needed) and
  evaluate its output signal values based on its
  current input signal values and states.
• Repeat step 2 until either all signal values have
  been resolved or no more signal values can be
  resolved.
• The set of all resolved signal values is called
  the least fixed point of the model at time t.
• Advance to another time t, repeat step 1.

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How to Advance Time?

• At any time, ask all blocks to post the next
  super-dense time they are expecting to produce
  outputs.
• Advance time to the earliest posted time, discard
  all remaining times.
• A block is responsible for posting the right time
  and keeping a record of when it will produce
  outputs.
• Optimization for DE and hybrid system models
• Global event queue may be used to keep a record
  of all previously posted times when are confirmed
  to have outputs generated, so that to avoid
  duplicate postings of the same future time.

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Proof of Concept

• A distributed version of Newtons Cradle model

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Summary

• Developed a rigorous operational semantics that
  leverages principles from synchronous/reactive
  (SR) languages.
• Applied this operational semantics to
  discrete-event (DE), continuous-time (CT), SR,
  finite-state machine (FSM), and hybrid system (CT
  DE FSM) models.