The Operational Semantics of Hybrid Systems

Transcription

1 The Operaional Seanics of Hybrid Syses Edward A. Lee Professor, Chair of EE, and Associae Chair of EECS, UC Berkeley Wih conribuions fro: Ada Caaldo, Jie Liu, Xiaojun Liu, Elefherios Masikoudis, and Haiyang Zheng Cener for Hybrid and Ebedded Sofware Syses Invied Plenary Talk Hybrid Syses: Copuaion and Conrol (HSCC) Zurich, Swizerland, March 9, 25 The Preise Hybrid Syses can be hough of as execuable progras. In his case, hey need o be given an execuable seanics. Lee, Berkeley: 2

2 Ouline Signals wih disconinuiies Ideal solver seanics Choosing sep sizes Discree phase of execuion Miscellanenous issues Enabling vs. riggering guards Order of reacions o siulaneous evens Nondeerinisic sae achines Sapling disconinuous signals Zeno behaviors Lee, Berkeley: 3 A Hybrid Syses Exaple Consider wo asses on springs which, when hey collide, will sick ogeher wih a decaying sickiness unil he force of he springs pulls he apar again. Lee, Berkeley: 4

3 Modal Models The Masses acor refines o a sae achine wih wo saes, Separae and Togeher. The ransiions have guards and rese aps. Lee, Berkeley: 5 Mode Refineens Each sae has a refineen ha gives he behavior of he odal odel while in ha sae. Lee, Berkeley: 6

4 Modeling Dynaics wihin he Separae Mode Dynaics while separae: Equivalenly: Lee, Berkeley: 7 Mode Refineens (2) In he Togeher ode, he dynaics is ha of a single ass and wo springs. Lee, Berkeley: 8

5 Modeling Dynaics wihin he Togeher Mode Dynaics while ogeher: Lee, Berkeley: 9 Iplied in he Maheaical Forulaion: Coninuous-Tie Signals The usual forulaion of he signals of ineres is a funcion fro he ie line T (a conneced subse of he reals) o he reals: Such signals are coninuous a T if (e.g.): Lee, Berkeley:

6 Piecewise Coninuous Signals In hybrid syses of ineres, signals have disconinuiies. Piecewise coninuous signals are coninuous a all T \ D where D T is a discree se. A se D wih an order relaion is a discree se if here exiss an order ebedding o he inegers). Lee, Berkeley: Operaional Seanics of Hybrid Syses A copuer execuion of a hybrid syse is consrained o provide values on a discree se: Given his consrain, choosing T as he doain of hese funcions is an unforunae choice. I akes i ipossible o unabiguously represen disconinuiies. Lee, Berkeley: 2

7 Definiion: Coninuously Evolving Signal Change he doain of he funcion: Where T is a conneced subse of he reals and se of naural nubers. is he A each ie T, he signal x has a sequence of values. Where he signal is coninuous, all he values are he sae. Where is disconinuous, i has uliple values. Lee, Berkeley: 3 Sipler Exaple: Hyseresis This odel shows he use of a wo-sae FSM o odel hyseresis. Seanically, he oupu of he ModalModel block is disconinuous. If ransiions ake zero ie, his is odeled as a signal ha has wo values a he sae ie, and in a paricular order. Lee, Berkeley: 4

8 Signals Mus Have Muliple Values a he Tie of a Disconinuiy Disconinuiies need o be seanically disinguishable fro rapid coninuous changes. Lee, Berkeley: 5 Iniial and Final Value Signals A signal has no chaering Zeno condiion if here is an ineger >such ha A non-chaering signal has a corresponding final value signal, where I also has an iniial value signal where Lee, Berkeley: 6

9 Piecewise Coninuous Signals A piecewise coninuous signal is a non-chaering signal where The iniial signal x i is coninuous on he lef, The final signal x f is coninuous on he righ, and The signal x has only one value a all T \ D where D T is a discree se. Lee, Berkeley: 7 Ouline Signals wih disconinuiies Ideal solver seanics Choosing sep sizes Discree phase of execuion Miscellanenous issues Enabling vs. riggering guards Order of reacions o siulaneous evens Nondeerinisic sae achines Sapling disconinuous signals Zeno behaviors Lee, Berkeley: 8

10 Discree Trace: Wha i Means o Execue a Hybrid Syse Le D' T be a discree se ha includes a leas he iniial ie and he ies of all disconinuiies. A discree race of he signal x is a se: An execuion of a hybrid syse is he consrucion of a discree race: s Lee, Berkeley: 9 Ideal Solver Seanics [Liu and Lee, HSCC 23] In he ideal solver seanics, he ODE governing he hybrid syse has a unique soluion for all inervals [ i, i+ ) for each neighboring i < i+ D'. The discree race loses nohing by no represening values wihin hese inervals. Alhough an idealizaion, his is no far feched. The spring asses exaple, for insance, confors wih he assupions and can be execued by an ideal solver s Lee, Berkeley: 2

11 Modeling Coninuous Dynaics wih Discree Traces A basic coninuousie odel describes an ordinary differenial equaion (ODE). Lee, Berkeley: 2 Srucure of he Model of Coninuous Dynaics x A basic coninuousie odel describes an ordinary differenial equaion (ODE). f ( x(, x &( = f ( x(, x( = x( ) + ( τ ) dτ Lee, Berkeley: 22

12 Absraced Srucure of he Model of Coninuous Dynaics Beween disconinuiies, he sae rajecory is odeled as a vecor funcion of ie, x : T n R T = [, ) R f ( x(, x &( = f ( x(, f : R T R x( = x( ) + ( τ ) dτ The key o he ideal solver seanics is ha coninuiy and local Lipschiz condiions on f are sufficien o ensure uniqueness of he soluion over a sufficienly sall inerval of ie. x Lee, Berkeley: 23 Ouline Signals wih disconinuiies Ideal solver seanics Choosing sep sizes Discree phase of execuion Miscellanenous issues Enabling vs. riggering guards Order of reacions o siulaneous evens Nondeerinisic sae achines Sapling disconinuous signals Zeno behaviors Lee, Berkeley: 24

13 Poins on he Tie Line ha Mus Be Included in a Discree Trace Predicable breakpoins Can be regisered in advance wih he solver Unpredicable breakpoins Known afer hey have been issed Poins ha ake he sep size sufficienly sall Dependen on error esiaion in he solver Require backracking Lee, Berkeley: 25 E.g. Runge-Kua 2-3 Solver (RK2-3) Given x( n ) and a ie increen h, calculae ( K = f ( x( ), ) n ) n n esiae of K = f ( x( n) +.5hK, n +.5h) ( n +.5h) esiae of K2 = f ( x( n) +.75hK, n +.75h) ( n +.75h) hen le n+ = x( ) = x( ) + (2 / 9) hk + (3/ 9) hk + (4 / 9) hk n+ n + h n Noe ha his requires hree evaluaions of f a hree differen ies wih hree differen inpus. 2 Lee, Berkeley: 26

14 Operaional Requireens In a sofware syse, he blue box below can be specified by a progra ha, given x( and calculaes f (x(, ). Bu his requires ha he progra be funcional (have no side effecs). f ( x(, x( = x( ) + ( τ ) dτ x x &( = f ( x(, f : R T R Lee, Berkeley: 27 Adjusing he Tie Seps For ie sep given by n + = n + K3 = f ( x( n+ ), n+ ) ε = h(( 5 / 72) K + (/2) K h, le + (/ 9) K + ( / 8) K If ε is less han he error olerance e, hen he sep is deeed successful and he nex ie sep is esiaed a: h = 3.8 e / ε If ε is greaer han he error olerance, hen he ie sep h is reduced and he whole hing is ried again. 2 3 ) Lee, Berkeley: 28

15 Exaining This Copuaionally f ( x(, x( = x( ) + ( τ ) dτ x A each discree ie n, given a ie increen n+ = n + h, we can esiae x( n+ ) by repeaedly evaluaing f wih differen values for he arguens. We ay hen decide ha h is oo large and reduce i and redo he process. Lee, Berkeley: 29 How General Is This Model? Does i handle: Syses wihou feedback? yes Exernal inpus? yes Sae achines? no The odel iself as a funcion? no x f ( x(, x( = x( ) + ( τ ) dτ x x &( = f ( x(, Lee, Berkeley: 3

16 How General Is This Model? Does i handle: Syses wihou feedback? yes Exernal inpus? yes Sae achines? no The odel iself as a funcion? no x f x( = x() + ( τ ) dτ x &( = f ( x(, Lee, Berkeley: 3 How General Is This Model? Does i handle: Syses wihou feedback? yes Exernal inpus? yes Sae achines? no The odel iself as a funcion? no f u g x x( = x() + ( τ ) dτ x &( = f ( x(, = g( u(, x(, Lee, Berkeley: 32

17 How General Is This Model? Does i handle: Syses wihou feedback? yes Exernal inpus? yes Sae achines? no, no iediaely The odel iself as a funcion? no x x( = x() + ( τ ) dτ ( f ( x(, Lee, Berkeley: 33 Acors wih Sae Mus Expose ha Sae Basic acor wih firing: Saeful acor: s S s 2 S S = [ T R] f : R T R T, s2( = f ( s(, s S s 2 S S = [ T N R] f : Σ R g : Σ R T R T Σ sae space (, n) T N, s2 (, n) =? The new funcion f gives oupus in ers of inpus and he curren sae. The funcion g updaes he sae a he specified ie. Lee, Berkeley: 34

18 Saeful Acors Suppor Unpredicable Breakpoins and Sep Size Adapaion s S s 2 S S = [ T N R] f : Σ R g : Σ R T R T Σ A each T he calculaion of he oupu given he inpu is separaed fro he calculaion of he new sae. Thus, he sae does no need o updaed unil afer he sep size has been decided upon. In fac, a variable sep size solver relies on his, since any of several inegraion calculaions ay resul in refineen of he sep size because he error is oo large. Lee, Berkeley: 35 How General Is This Model? Does i handle: Syses wihou feedback? yes Exernal inpus? yes Sae achines? yes, wih saeful acors The odel iself as a funcion? yes, bu be careful! f ( x(, x x( = x( ) + ( τ ) dτ x( = x( ) + ( τ ) dτ x Lee, Berkeley: 36

19 Why do we Care? Coposiionaliy Haiyang Zheng noiced ha earlier versions of HyVisual did no exhibi coposiional behavior. A correc resul A designer expecs cerain invarians: ransforaions of a odel ha do no change behavior. Resuls are calculaed wih he RK 2-3 solver. An incorrec resul Lee, Berkeley: 37 Why is Coposiionaliy Difficul o Achieve? In general, he behavior of he inside syse us be given by funcions of for: f : Σ R g : Σ R T R T Σ f ( x(, x x( = x( ) + ( τ ) dτ x( = x( ) + ( τ ) dτ x To ake his work, he sae of he solver us be par of he sae space Σ of he coposie acor! Lee, Berkeley: 38

20 Coposiional Execuion Requires ha Solvers Expose Deails An RK 2-3 solver evaluaes signal values a inerediae poins in ie ha do no ruly qualify as a sep. Given wo RK 2-3 solvers in a hierarchy, if hey do no cooperae on his, hen he behavior is alered by he hierarchy. The HyVisual Soluion: Solvers ha are separaed in he hierarchy by a os a Modal Model cooperae if hey are he sae ype of solver. This is coposiional, bu This also allows heerogeneous ixures of solvers. Lee, Berkeley: 39 Ouline Signals wih disconinuiies Ideal solver seanics Choosing sep sizes Discree phase of execuion Miscellanenous issues Enabling vs. riggering guards Order of reacions o siulaneous evens Nondeerinisic sae achines Sapling disconinuous signals Zeno behaviors Lee, Berkeley: 4

21 Transien Saes A Useful Model for Sofware If an ougoing guard is rue upon enering a sae, hen he ie spen in ha sae is idenically zero. This is called a ransien sae. Lee, Berkeley: 4 Transien Values Inegrae o Zero Transien values do no affec he inegral of he signal, as expeced. Lee, Berkeley: 42

22 Conras wih Siulink/Saeflow In Siulink seanics, a signal can only have one value a a given ie. Consequenly, Siulink inroduces solver-dependen behavior. The siulaor engine of Siulink inroduces a non-zero delay o consecuive ransiions. Transien Saes Lee, Berkeley: 43 Discree Phase of Execuion s S s 2 S S = [ T N R] f : Σ R g : Σ R T R T Σ A each T he oupu is a sequence of one or ore values where given he curren sae σ ( Σand he inpu s ( we evaluae he procedure s (,) = f ( σ (, s (, 2 σ ( = g( σ (, s (, s (,) = 2 f ( σ (, s (, σ ( = g( σ (, s (, 2... unil he sae no longer changes. We use he final sae on any evaluaion a laer ies. Coi o sep here Lee, Berkeley: 44

23 Ouline Signals wih disconinuiies Ideal solver seanics Choosing sep sizes Discree phase of execuion Miscellanenous issues Enabling vs. riggering guards Order of reacions o siulaneous evens Nondeerinisic sae achines Sapling disconinuous signals Zeno behaviors Lee, Berkeley: 45 Issue : Enabling vs. Triggering Guards In Modal Models, guards on could have eiher of wo seanic inerpreaions: enabling or riggering. If only enabling seanics are provided, hen i becoes nearly ipossible o give odels whose behavior does no depend on he sep-size choices of he solver. HyVisual uses riggering seanics. Enabling seanics can be realized wih an explici Mone Carlo odel. Lee, Berkeley: 46

24 Issue 2: Order of Reacion o Siulaneous Evens Given an even fro he even source, which of hese should reac firs? Nondeerinisic? Daa precedences? Seanics of a signal: s : T N R In HyVisual, every coninuousie signal has a value a (, ) for any T. This yields deerinisic execuion of he above odel. Siulink/Saeflow and HyVisual declare his o be deerinisic, based on daa precedences. Acor execues before Acor2. Soe foral hybrid syses languages declare his o be nondeerinisic. We believe his is he wrong choice. Lee, Berkeley: 47 Issue 3: Nondeerinisic Sae Machines HyVisual suppors explici Mone Carlo odels of nondeerinis. Alhough his can be done in principle, HyVisual does no suppor his sor of nondeerinis. Wha execuion race should i give? A a ie when he even source yields a posiive nuber, boh ransiions are enabled. Lee, Berkeley: 48

25 Issue 4: Sapling Disconinuous Signals Coninuous signal wih saple ies chosen by he solver: Discree resul of sapling: Saples us be deerinisically aken a - or +. Our choice is -, inspired by hardware seup ies. Noe ha in HyVisual, unlike Siulink, discree signals have no value excep a discree poins. Lee, Berkeley: 49 Issue 5: Zeno Condiions Zeno behavior is a propery of he discree evens in a syse, no a propery of is coninuous dynaics. The coninuous dynaics erely deerine he ie beween evens. Lee, Berkeley: 5

26 Zeno Behavior Can Be Deal Wih (alos Enirely in Discree Evens. Le he se of all signals be S = [T N V ] where V is a se of values. Le an acor be a funcion F : S n S. Wha are he consrains on such funcions such ha:. Coposiions of acors are deerinae. 2. Feedback coposiions have a eaning. 3. We can rule ou Zeno behavior. A sufficien condiion is ha every feedback loop have a lower bound on is ie delay. See [Lee 999] for a review of his resul, based on he Canor Meric. Lee, Berkeley: 5 Observaions If here is a lower bound on he sep size: All signals are discree here is an order ebedding o he naural nubers Inegraors wih he RK2-3 solver are dela causal, so soluion wih feedback is unique no Zeno in discreized seps bu lower bound on he sep size iplies inaccuracies Inegraors wih soe ehods (e.g. rapeziodal rule) are no dela causal, nor even sricly causal, so we have no assurance of a unique soluion in feedback syses. Lee, Berkeley: 52

27 Suary Signals us be able o have uliple values a a ie. Acors us separae reacions o inpus fro sae updaes Suppors even deecion Allows ieraive sep-size adjusen Coposiionaliy Need o be able o ix solvers Need o be able o add hierarchy wihou changing behavior Many deail issues in designing execuable hybrid syses: Guards should rigger raher han enable ransiions. Precedence analysis is essenial. Nondeerinis is easily added wih Mone Carlo ehods Sapling a disconinuiies needs o be well-defined. Zeno condiions are a discree even phenoenon Lee, Berkeley: 53 Open Source Sofware: HyVisual Execuable Hybrid Syse Modeling Buil on Poley II HyVisual 5.- alpha was released in March, 25. Lee, Berkeley: 54