Hybrid Automataand the

Continuous LifeNathan Sorenson

sparkfund.co@takeoutweight

• 2008 Surrey

• 2011 Downtown Vancouver

• 2015 East Van• 1996 Hybrid Automata

• thermostats

• communication networks

• biochemical reactions• Hybrid Automata

• based on an Actual Thing

• handles edge cases

• nice properties compose

• deterministic logical concurrency

• hybrid-expression

Dw = -1

w

timeDw = +1

w

dw dtw'ẇ

time

w

Dw=-1 Dw=+1

w<2

w>8

w ≥1

w ≤ 9

“control mode”

“flow condition”

“jump condition”

“invariant

condition”

x

x

y

y

Dy=+1Dx=-1

Dy=0Dx=0

Dx=0.2×x

t1

xt2

Dx=0.2×x

t1

xt2

x := x+1

x := x -1

xt := xt-δ -1

x=0.003×x

t1

xt2

x := x+1

x := x -1

ΔD

Hybrid Automata Features• motion in a control mode via derivatives• nondeterministic jumps (not= “random”)• interaction between variables• we can mutate variables via jumps• time can be continuous or discrete

pcba

$110

$108

$112

Dp=-1Da=+1 Db=+1

Dp=-1

a ≥108

Dc=+1Dp=-1

b ≥110 c ≥112

0

p ≤ 0

p

cba

Dc=2/6Dp=-3/6

Db=1/6Da=3/6

Dp=-6/6

Db=1/6Dc=2/6 Dc=2/6

Dp=-2/6

0

p

cb

a

Da=2/6Dp=-2/6

Da=2/6Dp=-3/6

Db=1/6Da=2/6Dp=-6/6

Db=1/6Dc=3/6

0

hybrid-expression• Algebraic Calculi for Hybrid Systems

Peter Hoefner 2009• Differential dynamic logic for hybrid

systems Platzer 2008

update-offsetupdate-flowuntil

sequencerepeatlazy-parstrict-par

Atomic expressions Combinators

x := x+1

update-offset : (Offset → Offset) → HExpr

(update-offset #(update % :x inc))

{:x 300} : Offset

!

update-flow : (Flow → Flow) → HExpr

(update-flow #(assoc % :p -1/6 :a 1/6))

{:p -1/6, :a 1/6, :x (he/* 0.3 :x)} : Flow

Dp=-1/6Da=1/6

Dp :=-1/6Da :=1/6

:x (he/* 0.3 :x)

!

w ≥ 8w <8

(until {:w (he/>= 8)})

until : OffsetPredicate → HExpr

A B

A⋅B

sequence : List<HExpr> → HExpr(sequence [A B])

A

Aω

repeat : HExpr → HExpr

(repeat A)

AB

lazy-par : List<HExpr> → HExpr

(lazy-par [A B])

∨

A B∨

||

!

AB

∧

A B∧

||

!

strict-par : List<HExpr> → HExpr

(strict-par [A B])

AB

(strict-par [A B])

A B∧

||

update-offsetupdate-flowuntil

sequencerepeatlazy-parstrict-par

Atomic expressions Combinators

p

cb

a

Da=2/6Dp=-2/6

Da=2/6Dp=-3/6

Db=1/6Da=2/6Dp=-6/6

Db=1/6Dc=3/6

0

a ≥10 b ≥12

(sequence [

(until {:p (he/<= 0)}) ])

a ≥10

Da=2/6Dp=-2/6

Da=2/6Dp=-3/6

Db=1/6Da=2/6Dp=-6/6

Db=1/6Dc=3/6

0

(update-flow (fn [fl] (-> fl (update :p #(- % 2/6)) (update :a #(+ % 2/6)))))

b ≥12

(update-flow [_] {:a 0 :p 0})

(sequence [(update-offset (fn [_] {:a 0 :b 0 :c 0 :p 50}))

])

(defn pay [v rate] (sequence [(update-flow (fn [fl] (-> fl (update :p #(- % rate)) (update v #(+ % rate))))) (until {:p (he/<= 0)}) (update-flow [_] {v 0 :p 0})]))

a ≥10

Da=2/6Dp=-2/6

Da=2/6Dp=-3/6

Db=1/6Da=2/6Dp=-6/6

Db=1/6

b ≥12

(strict-par [(pay :a 2/6) (sequence [(until {:a (he/>= 10)}) (pay :b 1/6)]) (sequence [(until {:b (he/>= 12)}) (pay :c 3/6)])])

Evaluating hybrid expressions

steady-state : HState → HState

HState = [Offset, Flow, other-stuff]

init : HExpr → HState

run : HState, Number → HState

(-> (steady-state))

(init waterfall)

FRPHybridAutomata

fast, accurate results

compose “processes”(not “time-varying values”)

• based on an Actual Thing

• handles edge cases

• nice properties compose

• deterministic logical concurrency

github.com/SparkFund• hybrid-expressio

n

Nathan Sorenson@takeoutweight