Towards component based design of hybrid systems
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Towards component based design of hybrid systems. W.Damm 1 , H. Dierks 3 , J. Oehlerking 4 , A. Pnueli 2. Structure of Presentation. Motivation and Industrial Context Hybrid Interface Specifications Component Based Design of Hybrid Systems: Assuring Safety and Stability Conclusion

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Towards component based design of hybrid systems

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Towards component based design of hybrid systems

W.Damm1, H. Dierks3, J. Oehlerking4, A. Pnueli2


Structure of Presentation

  • Motivation and Industrial Context

  • Hybrid Interface Specifications

  • ComponentBased Design of Hybrid Systems: AssuringSafetyandStability

  • Conclusion

    Thispresentationisbased on a publicationwhich will appear in the LNCS memorialvolumededicatedto

    Amir Pnueli


Motivation andindustrialcontext


The underlyingmathematics: hybrid automata


Answers requirement to decouple growth in number of functions from decoupling number of ECUs:

SW components of different functions can be allocated to one ECU

Allows SW components of one function to be distributed over multiple ECUs (to optimize overall architecture)

Components can correspond to different modes or subsystems of hybrid controllers

Induces distributed execution

Mode switching can cause task switching

Autosar Approach


Towardscomponentbased design of hybrid controllers

Can wepropose a component model for hybrid controllers

… supportingre-useofcomponents in multiple applicationcontexts?

  • Characterizingstabilityandsafetyproperties in specifiedenvironmentsthroughhybrid interfacespecifications

    … supportingincrementalconstructionof hybrid controllers

  • From a libraryofcontrollermodels

  • bycomposingcontrollersthroughtransitioncomposition

  • automaticverificationof hybrid interfacespecificationofcomposedsystemfrominterfacespecificationsofsubsystems

    … allowingtobridgethegapbetweenspecificationand design

  • Specificationmodelswithidealized time behaviour

  • Distributed implementationwithinducedimpurities such aslatencies in mode-switching


Hybrid Interface Specifications


Requirements on Hybrid Interface Specifications

  • Characterize plant regionsforwhichsafety and stabilityisguaranteed

  • Support compositionalreasoningforsafety and stability

  • Support transitionfromspecificationmodels to design

    • Specificationmodels

      • Focus on nominal behaviour

      • Assumeinstantenousobservability and controllability of plant

    • Design models

      • control-lawsbecometasks: supportactivation/suspension of components

      • provideexceptionhandlingadressingantitipatedrisksorfailures

      • caterfortask-switchinglatencies


The inner envelope design paradigm

Consider a safety property  given as conjunction of linear constraints. We identify an inner envelope o with the following properties

  • any only slightly perturbed trajectory originating in o stays there forever

  • whenever a sampled trajectory leaves o , then there is a time window of length at least  until  is violated when extrapolating the current dynamics even taking into account the specified worst-case dynamics for unmodelleddisturbances


Choose as entry condition an inner envelope of safesuch that all slightly disturbed trajectories originating in it will converge to (inner envelope) region of stability within specified bound

Similarly for stable

… and how we apply it

safe0

stable

stable0

safe

set-point


Raising alarms along bad trajectories

safe0

stable

stable0

safe

set-point

Combining Modes Safely


A ComponentLifecycle: threeroles

  • Controlunder nominal conditions

    • Ensure plant safety

    • Enforceconvergence of plant according to stabilityrequirements (asymptoticstability, drive plant intospecifiedregionwithingiven time bound)

  • Deviationsfromnonimalconditions:

    • Detectrisksforendangeringsafety and stability

    • Raisealarmearly to provideforsafetransition of control

  • Offeringhelp

    • Check forraisedalarms and offerhelpifcomponentspeccanadressdynamicscausingalarm


Approach

  • Componentsprovide

    • Inports:

      • To invoke nominal service

      • To offerhelp

      • To specify plant conditionsforwhichhelpcanbeoffered

    • Outports

      • To raisealarms

      • To characterize plant conditionscausingalarm

  • Componentscanraise multiple alarms

  • Conditionscausingalarmcandisappear


Specification of nominal behaviour

  • Stabilityrequirements

    • thissubsumesasymptoticstability

    • thecontrollerisrequired to meetthestabilityrequirementsunless an alarmisraised

  • Safetyrequirements

    • thecontrollerisrequired to meetthe plant safetyrequirementunless an alarmisraised


Being helpful: specification of inports

Is given by

where

  • cβsignals an incoming alarm

  • λβis the latest reaction time for granting acceptance

  • takeβsignals acceptance of alarm

  • startβis the verdict of the distributed alarm resolution protocol to become the hero

  • Mmmis the entry predicate required to be satisfied when control is transferred to the component over this port


Askingforhelp: specification of outports

Isgivenby

where

  • bαistheoutgoingalarmsignal

    isthe plant condition causingthealarm

  • μαisthe minimal persistency of thealarm

  • Δαisthedurationfollowingthealarmforwhichsafety and stabilityis still guaranteed

  • takeαsignalsthat at least onehelperisavailable

  • switchαsignalsdelegation of control to helper

  • Mmmoverapproximates plant state at switch time


  • Staticinterface

    • Data

    • Control


  • Inportspecifications

  • Outportspecifications


  • Stabilityrequirements

  • Assumptions

  • Promises


Hierarchical component based design

and verification


Hierarchicalconstruction of controllers

actuators

sensors

Plant


Sequentialcomposition of components

Pragmatics

  • All subsystems offer alternate ways of controlling same plant

  • Choice of subsystem dependent on current dynamics

  • if current subsystem is no longer able to ensure stability and safety objectives, a warning is raised using one of its exits

  • Control then either switches to other subsystem, or warning is passed to enclosing hierarchy level

  • Hence all subsystems share same static interface and safety and stability requirements relate to same equilibrium


Findingtheheroamong all offeringhelp

  • In a contextofincrementaldistributedcontrollerdesing, all of thesemightofferhelp

    • 5 neighbours on the same level of thehierarchy, but allocated on different Electronic Control Units

    • Some not yetknownfriend in a so-farunspecifiedenvironmentof thecomponent

  • Needdistributedagreementprotocol to ensureuniquetransfer of control

    • Wrapperforeachcomponent

    • Negotiateswithothercomponentswho will betheherousingprotocol on control-signals

      • Alarms, I cantakethis, Please do so, Activate, Suspend

      • Specifiedforeachinport


Real-timerequirementsfornegotiation

Negotiations must beclosedbeforesystembecomesunsafe

  • Criticalcomponentpromises to maintainsafety and stabilityforfixed time periodafterraisingalarm

  • takingintoaccountcostsforcontextswitches

  • Alarms mustensure minimal persistency to guaranteedistributedidenfication of helper

  • Helpers must provideoffer in given time window

  • Oncehelperisselected, it still takes tau time unitstoperformcontextswitch


Distributedagreementon heroes ...


Semantics of transitioncomposition

  • Let [[Ci]] denote hybrid automataexpressingthesemantics of subsystemCi .

  • Wedefinethesemantics [[C]] of thetransitioncomposition C = S(P,Q)(C1,...,Cn) as the parallel composition of hybrid automata

    • [[Ci]] representingthesemantics of itssubcomponents

    • HCpropagatingactivation and failures: itimplements

    • HQpropogatingcontrolsignalsfrominports: itimplements

    • HPimplementingdistributedidentification of hero


Distributedidentification of heroes ...

Automaton

codes in itsstateset

  • internallyraisedalarms

  • iffor such an alarmhelpersareavailable all such pairs (alarm, helper)

    Collects to this end all controlsignalsfromlocaloutports and controlsignals of localinports and externaloutportsbased on P-Port connection


CompositionalVerificationofstability - Approach

In a white-box viewwewouldconsiderthecomposedLyapunovfunctions V()

X | if in(Cj) thenVj(,X)

as a candidateLyapunovfunctionforthecomposedsystemandprove, thatthisfunctionisdecreasing

A keyingredient in thisproofis, thatcriticalitydoes not increase in modeswitching


Lyapunovfunctionsdemonstrateconvergence to equilibrium

  • Lyapunovfunctionprovidemeasuresofcriticalityofstatesoftheclosedloop H||P: redstatesarefarfrompointofequilibrium

  • Lyapunovfunctionsarewitnessesofstability: anytrajectoryoriginating in entry-regionofcontroller will convergetoequilibirum


Turning a hybrid automatainto a basiccomponentimplementation

  • Have to provideforactivation and suspension

  • Have to providewrappersupportingdistributedagreementprotocol

  • Leads to hybrid automatadefiningcomponentsemantics

  • Canverifywithautomatedverificationtechniquesthat hybrid automatameetscomponentinterfacespecifications

    • Nominal: safety and stability

    • Specifications of inports (partlyguaranteedbywrapperautomata)

    • Specifications of outports (partlyguaranteedbywrapperautomata)


Semantics of basiccomponents

Let

be a hybrid automataadmissableforcomponentspecification C and plant P. Wedefinethesemantics of theinducedcomponentimplementation I [[C(H)]] as the parallel composition of hybrid automata

with

  • H1allowingforchaoswhen I isnotactive

  • H2providingforactivation and suspension of H

  • H3supportingdistributedagreement on handling all alarms

  • Hβsupportingprotocolsforinports


Interface verificationofbasiccomponents (I)

Let

denotethe hybrid automatainducingthebasiccomponentimplementation, and considertheclosedloop H ||P .

Recall that a Lyapunovfunctionfor H||P is a function

meetingthefollowingrequirements


Verificationconditionsforbasiccomponents (1)

Nochattering – noimmediatealarms

wherereachreferstothelinear(!) closedloopdynamicsof H||P

Tools forestablishingverificationconditions:

-usingbarriercertificates/Lyapunovfunctions

-usingforwardreachabilityanalysistools such as PHAVER


Verificationconditionsforbasiccomponents (2)

  • Asymptoticstability

    • GeneratefamilyofLyapunovfunctionstoprovidemoreflexibilitywhencomposingsystems

    • for H||P

  • Time boundedconvergence

    • Weexploitthatany linear combinationof a Lyapunovfunctionsisagain a Lyapunovfunction

    • Letand


Verificationconditionsforbasiccomponents (3)

  • Exit conditionsareestablishedwithinescapeperiod

  • Promisesaremet

    Theorem

    If all verificationconditionsaresatisfied, then

    H||P satisfiesits hybrid interfacespecification


InductiveAssertions

As a basisforcompositionalgrey box verification, wemustprovidethefollowing „invariants“ inductively at theinterface of components

Additionally, parameterdependentconstantsforcomputingconvergencerates must bemadevisible


Conclusion and Future Work


Conclusion

  • Haveproposedtheoreticalfoundationforcomponentbaseddesign of hybrid controlsupportingcompositionalverification of nominal and exceptionhandlingrequirements

  • Verificationconditionsbothforbasic and composedsystemscanbedischargedautomatically

  • Future work

    • Extensions to parallel composition

    • Bridgingthegapbetweenidealized plant models and physicalplants


Thanks, Amir


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