EECE 396-1 Hybrid and Embedded Systems: Computation

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EECE 396-1Hybrid and Embedded Systems
Computation

• T. John Koo, Ph.D.
• Institute for Software Integrated Systems
• Department of Electrical Engineering and Computer
  Science
• Vanderbilt University
• 300 Featheringill Hall
• March 16, 2004
• john.koo_at_vanderbilt.edu
• http//www.vuse.vanderbilt.edu/kootj

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Computational Tool Reachability
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Transition System
Initial set
Unsafe Set
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Hybrid Automaton
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Hybrid Automaton
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Reachable Sets
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Reachable Sets
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Reachable Sets
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Reachable Sets
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Reachable Sets
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Reachable Sets
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Reachability Problem
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Reachability Problem
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Reachability Problem
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Reachability Problem
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Reachability Algorithm
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Reachability Algorithm
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Reachability Algorithm
Unsafe!!
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Reachability Algorithm
Safe!!
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Reachability Algorithm
Keep iterating until when!?
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Reachability Algorithm
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Reachability Algorithm
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Reachability Algorithm
Keep iterating until when!?
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Deciability
Keep iterating until when!?
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Deciability
Keep iterating until when!?
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Computational tools

• Basic computation includes
• Set-theoretic operations Union, Intersection,
  Difference
• Reach set computations Postd, Postc, Pred, Prec
• Verification
• Safety Property
• Forward algorithm
• Backward algorithm
• Liveness Property
• Properties specified by Temporal Logics
• Ref Thomas A. Henzinger, The Symbolic Approach
  to Hybrid Systems, (CAV02), UC Berkeley.

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Computational tools

• d/dt Library contributed by Thao Dang
• System Dynamics
• Linear systems
• Affine systems
• Linear systems with bounded inputs
• Set Representation
• Convex sets
• Basic (approximate) computation includes
• Set-theoretic operations Union, Intersection,
  Difference
• Reach set computations Postd, Postc, Pred, Prec
• Verification
• Specifications written as Temporal Logic Formula
• Algorithms

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Computational tools

• Projects
• Temporal Logic specifications
• Algorithms derivation
• d/dt based computational tool
• Verification
• Synthesis
• DC-DC Converters
• Controller verification
• Controller synthesis

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Computational tools
Temporal Logic
Specification
Input
Algorithm
Data Structure
Set Operations
Reach Sets
Dynamics
Output
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Design ExampleDC-DC Converters
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Power Electronics

• Power electronics found in
• DC-DC converters
• Power supplies
• Electric machine drives
• Circuits can be defined as networks of
• Voltage and current sources (DC or AC)
• Linear elements (R, L, C)
• Semiconductors used as switches (diodes,
  transistors)

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Power Electronics

• Discrete dynamics
• N switches, (up to) 2N discrete states
• Only discrete inputs (switching) some discrete
  transitions under control, others not
• Continuous dynamics
• Linear or affine dynamics at each discrete state

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Power Electronics DC-DC Converters

• Have a DC supply (e.g. battery), but need a
  different DC voltage
• Different configurations depending on whether
  VinltVout or VingtVout
• Control switching to maintain Vout with changes
  in load (R), and Vin

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Two Output DC-DC Converter

• Want two DC output voltages
• Inductors are big and heavy, so only want to use
  one
• Similar to two tank problem

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Circuit Operation

• One and only one switch closed at any time
• Each switch state has a continuous dynamics

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Design Objective
iL ?, VoutA?, VoutB ?
iL?, VoutA?, VoutB?
iL ?, VoutA ?, VoutB?
Objective Regulate two output voltages and limit
current by switching between three discrete
states with continuous dynamics.
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Typical Circuit Analysis/Control

• Governing equations
• Time domain, steady state
• Energy balance
• System dynamics
• Discretization in time
• Switched quantity only sampled at discrete
  instants
• Assumes a fixed clock
• Averaging
• Switched quantity approximated by a moving
  average
• Assumes switching is much faster than system time
  constants
• Control
• Linearize with duty (?) as input
• Use classical control techniques

iL(t)
iL(t)
iLk
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Problem Formulation
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Design ExampleDC-DC ConvertersController
Synthesis - Feasibility
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Problem Formulation

• Parallel Composition of Hybrid Automata
• Given a collection of Modes and Edges, design
  Guards

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Problem Formulation Hybrid Automaton
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FormulationCapacitor Discharging Mode (q1)
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FormulationCapacitor Charging Mode (q2)
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Backward Reachable sets (qualitative) w q2 q1
q1
q2
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d/dt Calculations result (quantitative)w q2
q1
NOT FEASIBLE
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Backward Reachable sets (qualitative) w q1
q2
q1
q2
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d/dt Calculations result (quantitative)w q1
q2
FEASIBLE
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Design ExampleDC-DC ConvertersController
Synthesis Switching Surfaces
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Switching Surface (Guard) Go Forward!w q1
q2
q1
q1
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Design ExampleDC-DC ConvertersController
Synthesis Simulation
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Problem Formulation

• Parallel Composition of Hybrid Automata
• Given a collection of Modes and Edges, design
  Guards

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Semi-Analytic Calculation of Switching Time
tsw0.158 ms
tsw0.174 ms
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End