From ec36d7a89baa3a9055f45666899fdb3bd74571cf Mon Sep 17 00:00:00 2001 From: Henri Saudubray Date: Tue, 9 Sep 2025 17:16:20 +0200 Subject: [PATCH] feat: presentation --- doc/data/0000001.csv | 301 ++ doc/data/10.csv | 301 ++ doc/data/ball.csv | 317 ++ doc/data/vdp.csv | 301 ++ doc/data/xsmall.csv | 9000 -------------------------------------- doc/pres.typ | 1393 +++--- doc/sources.bib | 19 + doc/zelus.sublime-syntax | 4 +- 8 files changed, 1826 insertions(+), 9810 deletions(-) create mode 100644 doc/data/0000001.csv create mode 100644 doc/data/10.csv create mode 100644 doc/data/ball.csv create mode 100644 doc/data/vdp.csv diff --git a/doc/data/0000001.csv b/doc/data/0000001.csv new file mode 100644 index 0000000..dc7702a --- /dev/null +++ b/doc/data/0000001.csv @@ -0,0 +1,301 @@ +0.,0.,2.953,1.,1. +0.1,0.292683750182,2.94619728416,1.09483761354,0.895170760251 +0.2,0.596874220104,3.16810443387,1.17873596328,0.781397224327 +0.3,0.929221643681,3.48280273578,1.25085678533,0.659816223251 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-9.997,1.78142035418,-0.161474723529 -9.998,1.78125885283,-0.161501350848 -9.999,1.78109732484,-0.161527991921 10.,1.7809357702,-0.161554646758 diff --git a/doc/pres.typ b/doc/pres.typ index f4b52bd..afabd02 100644 --- a/doc/pres.typ +++ b/doc/pres.typ @@ -1,860 +1,637 @@ + #import "@preview/cetz:0.3.2" #import "@preview/cetz-plot:0.1.1" #import "@preview/finite:0.4.1" #import "@preview/touying:0.6.1": * #import themes.simple: * -// TODO: clarify between -// `solver.step : time * 's -> (time * (time -> 'y) * zin option) * 's` -// and `sim.step : (time * (time -> 'a)) * 's -> (time * (time -> 'b)) * 's` +// Notes +// +// Shorten: too long on the explanation +// Instead, rely on a simple, two-axes presentation: We need +// - continuously described variables: IVP, ODEs +// - scheduling discrete-time signals with continuous ones: solution in simulink +// is zc events +// Need an introduction: what the problem is, what are we doing +// - Executable assertions: before any kind of formal methods, a programmer's +// tool +// - Explaining the simulation as succintly as possible +// +// Zélus: +// - Cite Simulink +// +// Discrete nodes: +// - remove some stuff +// - show signals +// - "redge" -> "rising_edge" +// - "true, false" -> "T, F" +// - cite Mealy +// - OCaml datatype +// - "Given i : I as input stream, simulation defines o : O [...]" +// +// Van der Pol: +// - show the math and oscillator +// +// Continuous nodes: +// - remove some stuff +// - the other way around: model first, low-level representation later (as for +// the discrete) +// - does not typecheck! +// - remind the audience of the vdp oscillator math +// - f_der, f_out +// +// Hybrid nodes: +// - the other way around, too: model first, low-level representation later +// - f_zero +// +// Synthesis: new constructs for continuous time (hybrid, der, up...) +// +// Assertions: +// - Shorter, explain as a programmer's tool +// +// Now, my work: +// - define the runtime as precisely and succintly as possible; +// +// ODE/ZC solvers +// - Too "hand-wavy" #show: simple-theme.with( - aspect-ratio: "16-9", - config-common(show-bibliography-as-footnote: bibliography("sources.bib")), -) -#let cetz-canvas = touying-reducer.with( - reduce: cetz.canvas, - cover: cetz.draw.hide.with(bounds: true) -) -#let arrow(start, ..steps, stroke: black) = cetz.draw.line( - ..(steps.pos().fold((start,), (acc, step) => { - let ((.., (x, y)), (d, l)) = (acc, step) - let (nx, ny) = if d == "d" { - (x, y - l) - } else if d == "u" { - (x, y + l) - } else if d == "l" { - (x - l, y) - } else if d == "r" { - (x + l, y) - } else { - (x, y) - } - acc.push((nx, ny)) - acc - })), - mark: (end: "straight"), - stroke: stroke -) -#let (d, u, l, r) = ("d", "u", "l", "r") -#let box(start, dim, double: false, content: none, ..rest) = { - let (x, y) = start - let (dx, dy) = dim - cetz.draw.rect((x, y), (x + dx, y + dy), ..rest) - if double { - cetz.draw.rect((x + 0.1, y + 0.1), (x + dx - 0.1, y + dy - 0.1), ..rest) - } - if content != none { - cetz.draw.content((x + dx / 2, y + dy / 2), content) - } -} + config-common(show-bibliography-as-footnote: bibliography("sources.bib"))) #set raw(syntaxes: "zelus.sublime-syntax") #show raw: set text(font: "CaskaydiaCove NF") +#let zelus = smallcaps[Zélus] +#let lustre = smallcaps[Lustre] +#let simulink = smallcaps[Simulink] +#let ocaml = smallcaps[OCaml] +#let Time = math.italic[Time] -= Executing Hybrid Systems +// 20 mins presentation - 10 mins questions + += Internship Defense --- MPRI M2 \ Hybrid System Models with Transparent Assertions -Henri Saudubray -#footnote(link("https://codeberg.org/17maiga/hsim"), numbering: "*") +Henri #smallcaps[Saudubray]#footnote( + [All code accessible at #link("https://codeberg.org/17maiga/hsim").], + numbering: "*" +), supervised by Marc #smallcaps[Pouzet], #smallcaps[Inria PARKAS] -22/05/2025 +10/09/2025 += Context: #zelus and Hybrid System Models -= Hybrid Systems +== The #zelus Programming Language -== Hybrid systems, as seen in Zélus +- Language@cit:zelus_sync_lng_with_ode for description of hybrid system models +- Synchronous language kernel _à la_ #lustre@cit:lustre +- Extension to continuous-time behaviour +- Compilation to #ocaml code +- Execution using an off-the-shelf numerical solver@cit:sundials@cit:sundialsml -Computation alternates between _discrete_ and _continuous_ phases. -#pause #linebreak() -Use of an ODE solver to approximate continuous behaviour. -#pause #linebreak() -Side effects and state jumps occur in discrete steps triggered by -zero-crossings. +== Discrete-Time Systems -#meanwhile +Programs (nodes) are functions on streams -#align(horizon, { - columns(2, [ - ```zelus - let hybrid saw() = y where - rec der y = 1.0 - init 0.0 - reset z -> (0.0) - and z = up(y -. 1.0) - ``` - #align(center, { - import cetz: * - import cetz-plot.plot: * - canvas({ - plot( - size: (9, 4), axis-style: "school-book", legend: (7.25, 4.5), - x-tick-step: 1, x-label: none, y-tick-step: 1, y-label: none, - y-grid: true, y-min: -0.5, y-max: 1.5, x-min: 0, x-max: 4.5, - plot-style: (stroke: blue), { - for i in range(5) { - add(((i, 0), (i + 1, 1))) - } - add(((0, 0),), label: [`saw`]) - }) - }) - }) - ]) -}) - -== Hybrid assertions - -Assertions are themselves hybrid systems. -#pause #linebreak() -Adding assertions can change the final result, they are not _transparent_. -#pause #linebreak() -Solution: simulate assertions with their own solver. -#meanwhile - -#columns(2, [ - #align(horizon, [ - ```zelus - let hybrid saw() = y where - rec der y = 1.0 - init 0.0 - reset z -> (0.0) - and z = up(y -. 1.0) - and assert(0.0 <= y <= 1.0) - ``` - ]) - #colbreak() - #align(center + horizon, { - import cetz: * - canvas({ - import cetz.draw: * - box((0, 0), (8, 4.5)) - box((2, 0.25), (4, 1.5), content: ```zelus assert```) - box((2, 2.75), (4, 1.5), content: `...`) - arrow((-1, 3.5), (r, 3)) - arrow((6, 3.5), (r, 3)) - arrow((7, 3.5), (d, 1.25), (l, 6), (d, 1.25), (r, 1)) - content((-1.5, 3.5), [`()`]) - content((9.5, 3.5), [`y`]) - content((1, 5), [`saw`]) - }) - }) +#align(center + horizon, [ + ```zelus + let node rising_edge(x: bool) = + x -> (x && not(pre(x))) + ``` + #linebreak() + #table( + columns: (auto, auto, auto, auto, auto, auto, auto), + align: center + horizon, stroke: (x, y) => ( + left: if x == 0 { 0pt } else { 1pt }, + top: if y == 0 { 0pt } else { 1pt }, + ), + `x`, $F$, $T$, $T$, $F$, $T$, [...], + ```zelus pre(x)```, $bot$, $F$, $T$, $T$, $F$, [...], + ```zelus x && not(pre(x))```, $bot$, $T$, $F$, $F$, $T$, [...], + ```zelus x -> (x && not(pre(x)))```, $F$, $T$, $F$, $F$, $T$, [...], + ) ]) -= Nodes +// Prononcer les suites, "front montant", `fby` ? -== Discrete nodes +#pagebreak() -Nodes take an additional reset parameter `'p` for their reset function: +Compiled down to Mealy machines@cit:mealy@cit:sync_based_codegen_hyb_sys_lng: +$ s_0 : S wide wide f_"step" : S times I -> O times S $ +#pause +With input $i : NN -> I$, simulation defines output $o : NN -> O$ +$ (o_n, s_(n+1)) = f_"step" (s_n, i_n) $ -#align(top)[ - #grid(columns: (3fr, 2fr), align: (left, left), - ```ocaml - type ('p, 'a, 'b) dnode = - DNode : { - state : 's; - step : 's -> 'a -> 'b * 's; - reset : 'p -> 's -> 's; - } -> ('p, 'a, 'b) dnode - ```, - cetz.canvas({ - import cetz.draw: * - box((0, 0), (4, 4), content: `DNode`) - arrow((-1, 2), (r, 1)) - arrow((4, 2), (r, 1)) - arrow((2, 5), (d, 1)) - content((-1.6, 2), `'a`) - content((5.6, 2), `'b`) - content((2, 5.6), `'p`) +== Physical Simulation in Discrete Time + +// Mention initial values in vdp + +The Van der Pol oscillator: +$ (d x)/(d t) = y wide (d y)/(d t) = mu(1 - x^2)y - x wide x(0) = y(0) = 1 $ +#pause +#align(center, cetz.canvas({ + cetz-plot.plot.plot(size: (20, 4), axis-style: "school-book", + x-tick-step: 10, y-tick-step: 5, x-label: $t$, y-label: none, + y-grid: "both", legend: (19, 4.5), { + let d = csv("data/vdp.csv") + let x = d.map(r => (r.at(0), r.at(1)).map(v => float(v.trim()))) + let y = d.map(r => (r.at(0), r.at(2)).map(v => float(v.trim()))) + cetz-plot.plot.add(x, line: "spline", label: $x$) + cetz-plot.plot.add(y, line: "spline", label: $y$) + }) +})) + +#pagebreak() + +#only((1,2))[ + ```zelus + let h = 0.001 + (* Euler integrators. *) + let node f_euler(x0, x') = x where + rec x = x0 -> pre(x + x' * h) + let node b_euler(x0, x') = x where + rec x = x0 -> pre(x) + x' * h + + (* Van der Pol oscillator. *) + let node vdp(μ) = (x, y) where + rec x = b_euler(1.0, y) + and y = f_euler(1.0, (μ * (1.0 - (x * x)) * y) - x) + ``` +] +#only(2)[ + #place(dx: 55%, dy: -70%, cetz.canvas({ + let d = csv("data/xsmall.csv") + let x = d.map(r => (r.at(0), r.at(1)).map(v => float(v.trim()))) + let y = d.map(r => (r.at(0), r.at(2)).map(v => float(v.trim()))) + cetz-plot.plot.plot(size: (10, 5), axis-style: "school-book", + x-tick-step: 10, y-tick-step: 5, x-grid: "both", y-grid: "both", + y-min: -8, y-max: 8, legend: (9, 5.5), y-label: none, x-label: none, { + cetz-plot.plot.add(x, label: $x$) + cetz-plot.plot.add(y, label: $y$) + }) + })) +] +#only(3)[ + ```zelus + let h = 0.1 + (* Euler integrators. *) + let node f_euler(x0, x') = x where + rec x = x0 -> pre(x + x' * h) + let node b_euler(x0, x') = x where + rec x = x0 -> pre(x) + x' * h + + (* Van der Pol oscillator. *) + let node vdp(μ) = (x, y) where + rec x = b_euler(1.0, y) + and y = f_euler(1.0, (μ * (1.0 - (x * x)) * y) - x) + ``` + #place(dx: 55%, dy: -70%, cetz.canvas({ + let d = csv("data/middle.csv") + let x = d.map(r => (r.at(0), r.at(1)).map(v => float(v.trim()))) + let y = d.map(r => (r.at(0), r.at(2)).map(v => float(v.trim()))) + cetz-plot.plot.plot(size: (10, 5), axis-style: "school-book", + x-tick-step: 10, y-tick-step: 5, x-grid: "both", y-grid: "both", + y-min: -8, y-max: 8, legend: (9, 5.5), y-label: none, x-label: none, { + cetz-plot.plot.add(x, label: $x$) + cetz-plot.plot.add(y, label: $y$) + }) })) ] -== Hybrid nodes +== Continuous-Time Systems -Hybrid nodes are a bit more complex: -#block([ - ```ocaml - type ('p, 'a, 'b, 'y, 'yder, 'zin, 'zout) hnode = - HNode : { - state : 's; - step : 's -> 'a -> 'b * 's; - reset : 'p -> 's -> 's; - fder : 's -> 'a -> 'y -> 'yder; - fzer : 's -> 'a -> 'y -> 'zout; - fout : 's -> 'a -> 'y -> 'b; - (* ... *) - } -> ('p, 'a, 'b, 'y, 'yder, 'zin, 'zout) hnode - ``` - #pause - #place(left + top, dx: 75%, dy: 36%, block(grid( - align: left + horizon, columns: (1fr, 10fr), - move(scale(y: 200%, $}$), dy: -2%), [Discrete behaviour] - ), width: 40%)) - #place(left + top, dx: 75%, dy: 62.5%, block(grid( - align: left + horizon, columns: (1fr, 10fr), - move(scale(y: 300%, $}$), dy: -2%), [Continuous behaviour] - ), width: 40%)) -]) +Extension to continuous time. \ #pause +New constructs: ```zelus hybrid```, ```zelus der```, ```zelus init``` -= Solvers are nodes +$ (d x)/(d t) = y wide (d y)/(d t) = mu(1 - x^2)y - x wide x(0) = y(0) = 1 $ -== ODE solvers - -#slide(repeat: 3, self => [ - #let (uncover, only) = utils.methods(self) - - ODE solvers are discrete nodes producing streams of functions defined on - successive intervals: - $(h: bb(R)_+) -->^D (h': bb(R)_+) times (u: [0,h'] -> bb(V))$. - - #linebreak() - #grid(columns: (1fr, 1fr), align: (center, left), cetz.canvas({ - import cetz.draw: * - box((0, 0), (4, 4), content: `csolver`) - arrow((-1, 2), (r, 1)) - arrow((4, 3), (r, 1)) - arrow((4, 1), (r, 1)) - content((-1.5, 2), $h$) - content((5.5, 3), $h'$) - content((5.5, 1), $u$) - arrow((2, 5), (d, 1)) - content((2, 5.6), "IVP") - }), align(horizon, cetz-canvas({ - import cetz-plot.plot: * - only(1, plot( - size: (10, 4.5), axis-style: "school-book", legend: (9, 4.5), - y-tick-step: none, y-label: none, y-min: 0, y-max: 6, - x-tick-step: none, x-label: none, x-min: 0, x-max: 5, - x-ticks: ((5, $h$), (2, $h'$)), { - add(domain: (0, 2), t => calc.sin(t) + t, label: $u$) - })) - only(2, plot( - size: (10, 4.5), axis-style: "school-book", legend: (9, 4.5), - y-tick-step: none, y-label: none, y-min: 0, y-max: 6, - x-tick-step: none, x-label: none, x-min: 0, x-max: 5, - x-ticks: ((5, $h$), (2, ""), (4, $h'$)), { - add(domain: (2, 4), t => calc.sin(t) + t, label: $u$) - add(domain: (0, 2), t => calc.sin(t) + t) - })) - only(3, plot( - size: (10, 4.5), axis-style: "school-book", legend: (9, 4.5), - y-tick-step: none, y-label: none, y-min: 0, y-max: 6, - x-tick-step: none, x-label: none, x-min: 0, x-max: 5, - x-ticks: ((2, ""), (4, ""), (5, $h', h$)), { - add(domain: (4, 5), t => calc.sin(t) + t, label: $u$) - add(domain: (0, 4), t => calc.sin(t) + t) - })) - }))) -]) - -== Zero-crossing solvers - -#slide(repeat: 3, self => [ - #let (uncover, only) = utils.methods(self) - - Zero-crossing solvers are discrete nodes too, looking for zero-crossings on - functions: - $(h: bb(R)_+) times (u: [0, h] -> bb(V)) -->^D - (h': bb(R)_+) times (z: bb(Z)?)$. - - #linebreak() - #grid(columns: (1fr, 1fr), align: (center, left), cetz.canvas({ - import cetz.draw: * - box((0, 0), (4, 4), content: `zsolver`) - arrow((-1, 1), (r, 1)) - arrow((-1, 3), (r, 1)) - arrow((4, 1), (r, 1)) - arrow((4, 3), (r, 1)) - arrow((2, 5), (d, 1)) - content((-1.5, 3), $h$) - content((-1.5, 1), $u$) - content((5.5, 3), $h'$) - content((5.5, 1), $z$) - content((2, 5.6), "ZC") - }), align(horizon, cetz-canvas({ - import cetz-plot.plot: * - only(1, plot( - size: (10, 4.5), axis-style: "school-book", legend: (9, 4.5), - y-tick-step: none, y-label: none, y-min: -2, y-max: 5, - x-tick-step: none, x-label: none, x-min: 0, x-max: 2, - x-ticks: ((2/3, $h', h$),), { - add(domain: (0, 2/3), t => t * t - 1, label: "u") - })) - only(2, plot( - size: (10, 4.5), axis-style: "school-book", legend: (9, 4.5), - y-tick-step: none, y-label: none, y-min: -2, y-max: 5, - x-tick-step: none, x-label: none, x-min: 0, x-max: 2, - x-ticks: ((1, $h'$), (4/3, $h$)), { - add(domain: (2/3, 4/3), t => t * t - 1, label: "u") - add(domain: (0, 2/3), t => t * t - 1) - })) - only(3, plot( - size: (10, 4.5), axis-style: "school-book", legend: (9, 4.5), - y-tick-step: none, y-label: none, y-min: -2, y-max: 5, - x-tick-step: none, x-label: none, x-min: 0, x-max: 2, - x-ticks: ((2, $h', h$),), { - add(domain: (1, 2), t => t * t - 1, label: "u") - add(domain: (0, 1), t => t * t - 1) - })) - only("1, 3", cetz.draw.content((2, 4), $z = bot$)) - only("2", cetz.draw.content((2, 3.9), $z != bot$)) - }))) -]) - -== A full solver - -Combine an ODE solver `csolver` with a zero-crossing solver `zsolver`: - -#linebreak() - -#align(center, { - import cetz: * - canvas({ - import cetz.draw: * - box((0, 0), (12, 4)) - content((1.5, 4.5), `solver`) - box((1, 0.5), (4, 3), content: `csolver`) - box((7, 1.5), (4, 2), content: `zsolver`) - arrow((-1, 2), (r, 2)) - arrow((5, 3), (r, 2)) - arrow((5, 1), (r, 8)) - arrow((6, 1), (u, 1), ("r", 1)) - arrow((11, 2), (r, 2)) - arrow((11, 3), (r, 2)) - content((-1.5, 2), [$h$]) - content((6, 3.5), [$h'$]) - content((6, 0.5), [$u$]) - content((13.5, 1), [$u$]) - content((13.75, 2), [$h''$]) - content((13.5, 3), [$z$]) - }) -}) - -= Signals - -== Semantics - -Two semantics are often used for hybrid systems: -- Superdense semantics @opsem_lee_zheng: - - signals are functions from $bb(R) times bb(N) -> bb(V)$; - - discrete instants are ordered lexicographically -- Non-standard semantics @ns_sem_benveniste_bourke_caillaud_pouzet: - - signals are functions from the hyperreals $""^*bb(R) -> bb(V)$; - - discrete instants are ordered using infinitesimals - -== Optional values - -The solver does not always reach the requested horizon in a single step. -#linebreak() -We _wait_ for it to finish computing: - -#align(center, { - import cetz: * - canvas({ - import cetz.draw: * - content((0, 1.5), "Input:") - content((0, 0), "Output:") - content((3.5, 1.5), $a_0,$) - content((3.5, 0), $bot,$) - content((5, 1.5), $bot,$) - content((5, 0), $b_0,$) - content((6.5, 1.5), $bot,$) - content((6.5, 0), $b_1,$) - content((8, 1.5), $bot,$) - content((8, 0), $b_2,$) - content((9.5, 1.5), $bot,$) - content((9.5, 0), $bot,$) - content((11, 1.5), $a_2,$) - content((11, 0), $bot,$) - content((12.5, 1.5), $bot,$) - content((12.5, 0), $b_3,$) - content((14, 1.5), $bot,$) - content((14, 0), $b_4,$) - content((15.5, 1.5), $...$) - content((15.5, 0), $...$) - }) -}) - -This is represented by optional values: - -$ "Signal"(alpha) = "Stream"("Optional"((h:bb(R)_+) times ([0,h] -> alpha))) $ - -== Discrete instants - -Discrete instants are represented by constant functions of length 0. Their -order is the order of apparition in the stream. - -== Example - -The stream -#align(center, ```ocaml -f = (1.0, λt -> t)::(0.0, λt -> 2.0)::(0.3, λt -> 3*t)::... -```) -defines the function $f : ""^*bb(R)_+ -> bb(R)$ -#grid(columns: (1fr, 1fr), align: (center, center), $ & f(t) = cases( - "st"(t)#footnote([ - $"st"(t)$ denotes the _standard part_ of $t$, that is, the closest real - number $r$ to $t$. - ]) & "if" t in [0, 1], - 2 & "if" t = 1 + partial, - 3 dot "st"(t) & "if" t in [1 + 2 partial, 1.3], - ... -) $, cetz.canvas({ - import cetz-plot.plot: * - plot( - size: (10, 3), axis-style: "school-book", - y-label: none, y-tick-step: none, - x-label: none, x-tick-step: none, x-ticks: (1, 2.3), x-grid: true, - { - add(domain: (0, 1), t => t) - add(((1, 2),), mark: "o", mark-size: 0.05, - mark-style: (stroke: blue, fill: blue)) - add(domain: (1, 1.3), t => 3 * t, style: (stroke: blue)) - } - ) -})) - -= Simulation - -== Simulation - -The simulation is itself a discrete node on signals: - -#align(center, cetz.canvas({ - import cetz.draw: * - box((0, 0), (12, 5), content: `???`) - arrow((-1, 2.5), (r, 1)) - arrow((12, 2.5), (r, 1)) - content((-3, 2.5), $"Signal"(alpha)$) - content((15, 2.5), $"Signal"(beta)$) - content((1, 5.5), `sim`) - box((0.5, 0.5), (4, 4), content: `model`) - box((7.5, 0.5), (4, 4), content: `solver`) -})) - -It runs according to one of two modes: _discrete_ and _continuous_. - -== (Re)initialization - -When receiving a new value, store it -#alternatives([...], [and reset the solver.]) -#align(center, cetz-canvas({ - import cetz.draw: * - - // sim - content((1, 6.5), `sim`) - box((0, 0), (14, 6)) - - // model - content((2.5, 5.25), `model`) - box((1, 1), (4, 3.75)) - - // model.fzer - box((1.25, 3), (3.5, 1.5), content: `fzer`) - - // model.fder - box((1.25, 1.25), (3.5, 1.5), content: `fder`) - - // sim state - box((10, 1.25), (3, 1.5), double: true, content: `state`) - - // (h, u) -> state - content((-2.25, 3), $(h, u)$) - arrow((-1, 3), (r, 1.5), (d, 2.5), (r, 5), (u, 1.5), (r, 4.5)) - - // sim -> bot - arrow((14, 3), (r, 1)) - content((15.5, 3), $bot$) - - (pause,) - - // computation - box((6, 3), (3, 1.5), content: `...`) - - // solver - box((10, 3), (3, 1.5), content: `solver`) - - // (h, u) -> computation - arrow((5.5, 2), (u, 1.5), (r, 0.5)) - - // fzer -> computation - arrow((4.75, 4), (r, 1.25)) - - // fder -> computation - arrow((4.75, 2), (r, 0.5), (u, 1.75), (r, 0.75)) - - // computation -> solver - arrow((9, 3.75), (r, 0.5), (u, 1.25), (r, 2), (d, 0.5)) - content((10.5, 5.5), `ivp + zc`) -})) - -== Discrete step - -#alternatives( - [Use the model's `step` function and the stored input.], - [Build a constant function out of the output value.], - [Update the running mode as needed.] -) -#align(center, cetz-canvas({ - import cetz.draw: * - // sim - content((1, 6.5), `sim`) - box((0, 0), (17, 6)) - - // sim state - box((1, 1), (3, 1.5), double: true, content: `state`) - - // model - content((6.5, 5.5), `model`) - box((5, 0.5), (7, 4.5)) - - // model step - box((7, 1), (3, 1.5), content: `step`) - - // model state - box((7, 2.75), (3, 1.5), double: true, content: `state`) - - // bot -> sim - arrow((-1, 3), (r, 1)) - content((-1.5, 3), $bot$) - - // sim.state -> model.step - arrow((4, 1.5), (r, 3)) - content((5.75, 1.25), $alpha$) - - // model.state -> model.step - arrow((7, 3.5), (l, 0.5), (d, 1.5), (r, 0.5)) - content((6, 2.875), $sigma$) - - (pause,) - - // constant signal - box((13, 1), (3, 1.5), content: `...`) - - // model.step -> model.state - arrow((10, 2), (r, 0.5), (u, 1.5), (l, 0.5)) - content((11, 2.875), $sigma$) - - // model.step -> ... - arrow((10, 1.5), (r, 3)) - content((11.25, 1), $beta$) - - // constant signal -> signal(beta) - arrow((16, 1.75), (r, 0.5), (u, 1.25), (r, 1.5)) - content((20, 3), $"Signal"(beta)$) - - (pause,) - - // computation for mode - box((1, 3.5), (3, 1.5), content: `...`) - - // model.step -> mode computations - arrow((10.5, 3.5), (u, 1), (l, 6.5)) - - // sim.state -> mod.computations - arrow((4, 2), (r, 0.5), (u, 2), (l, 0.5)) - - // mod.computations -> sim.state - arrow((1, 4.25), (l, 0.5), (d, 2.5), (r, 0.5)) -})) - -== Continuous step - -#alternatives( - [Call the solver with the current target horizon.], - [Use the current input and the result of the solver to build the output.], - [Update the model's continuous state with the new horizon.], - [Update the simulation mode in case of a zero-crossing.], -) - -#align(center, cetz-canvas({ - import cetz.draw: * - // sim - content((1, 6.5), `sim`) - box((0, 0), (17, 6)) - - // sim.state - box((1, 1), (3, 1.5), double: true, content: `state`) - - // bot -> sim - arrow((-1, 3), (r, 1)) - content((-1.5, 3), $bot$) - - // solver - content((6.5, 5.25), `solver`) - box((5, 0.75), (6, 4)) - - // solver.step - box((6.5, 1), (3, 1.5), content: `step`) - - // solver.state - box((6.5, 2.75), (3, 1.5), double: true, content: `state`) - - // solver.state -> solver.step - arrow((6.5, 3.5), (l, 0.5), (d, 1.5), (r, 0.5)) - content((5.5, 2.75), $sigma$) - - // sim.state -> solver.step - arrow((4, 1.5), (r, 2.5)) - - (pause,) - - // solver.step -> solver.state - arrow((9.5, 2.2), (r, 0.5), (u, 1.3), (l, 0.5)) - - // computation - box((13, 1), (3, 1.5), content: `...`) - - // solver.step -> computation - arrow((9.5, 1.6), (r, 3.5)) - arrow((9.5, 1.3), (r, 3.5)) - - // sim.state -> computation - arrow((4, 2), (r, 0.5), (d, 1.5), (r, 8), (u, 1.55), (r, 0.5)) - - // computation -> beta signal - arrow((16, 1.75), (r, 2.5)) - content((20.5, 1.75), $"Signal"(beta)$) - - (pause,) - - // model - box((12.5, 2.75), (4, 2)) - content((14, 5.25), `model`) - - // model.state - box((13, 3), (3, 1.5), double: true, content: `state`) - - // step -> model - arrow((12.25, 1.3), (u, 2.2), (r, 0.75)) - arrow((12, 1.6), (u, 2.4), (r, 1)) - - (pause,) - - // computation - box((1, 2.75), (3, 1.5), content: `...`) - - // solver.step -> computation - arrow((9.5, 1.9), (r, 1), (u, 2.6), (l, 6), (d, 1), (l, 0.5)) - - // computation -> solver.state - arrow((1, 3.5), (l, 0.5), (d, 1.75), (r, 0.5)) -})) -#pause -Is that it? - -= States - -== The state problem - -#slide(self => [ - #let (uncover, only) = utils.methods(self) - Some solvers modify their state in-place (e.g. #smallcaps("Sundials")). - #linebreak() - The function returned depends on this state. - #pause #linebreak() - It becomes invalid once we call the solver again. - - #meanwhile - #align(center, cetz-canvas({ - import cetz-plot.plot: * - plot( - size: (10, 4), axis-style: "school-book", - x-label: none, x-tick-step: none, x-max: 6, - y-label: none, y-tick-step: none, y-max: 6, - x-grid: true, x-ticks: ((4.5, ""),), - { - only(1, add(domain: (0, 4.5), t => calc.sin(t) + t)) - only(2, add(domain: (4.5, 6), t => calc.sin(t) + t)) - only(2, add(domain: (0, 4.5), t => calc.sin(t) + t)) - } - ) - })) -]) - -== Solver steps - -Most solvers start off with smaller steps, and increase the step length -progressively. - -#align(center, cetz.canvas({ - import cetz-plot.plot: * - plot( - size: (10, 4), axis-style: "school-book", x-label: none, x-tick-step: none, - x-ticks: ((0,""), (0.001,""), (0.01,""), (0.1,""), (0.2,""), (0.4,""), - (1,""), (1.9,""), (3.9, "")), x-grid: true, - y-label: none, y-tick-step: none, { - add(domain: (0, 6), t => calc.sin(t) + t) - } - ) -})) - -Combined with in-place state modifications, resets and composition, this could -hinder performance. - -== State copies - -Copying (part of) the state solves this issue. -#linebreak() -If the solver supports state copies, we can call it more than once per -simulation step. - -```ocaml -type ('p, 'a, 'b) dnode_c = - DNodeC : { - state : 's; - step : 's -> 'a -> 'b * 's; - reset : 'p -> 's -> 's; - copy : 's -> 's; - } -> ('p, 'a, 'b) dnode_c +```zelus +let hybrid vdp(μ) = (x, y) where + rec der x = y init 1.0 + and der y = (μ * (1.0 - (x * x)) * y) - x) init 1.0 ``` -== Accelerating the simulation +#pagebreak() -We can then concatenate functions obtained from successive continuous steps. -If no zero-crossing is found, there is no discontinuity. +Programs are compiled down to initial value problems (IVPs) +$ y_0 : Y wide f_"der" : Time times I times Y -> Y' + wide f_"out" : Time times I times Y -> O $ +#pause +Use a numerical solver to compute an approximation of $y : Time -> Y$ +$ y(0) = y_0 wide (d y)/(d t)(t) = f_"der" (t, i(t), y(t)) $ +#pause +With input $i : Time -> I$, simulation defines output +$o : Time -> O$ +$ o(t) = f_"out" (t, i(t), y(t)) $ -#linebreak() +== Hybrid Systems -#grid(align: center + horizon, columns: (10fr, 1fr, 10fr), - cetz.canvas({ - import cetz-plot.plot: * - plot( - size: (8, 4), axis-style: "school-book", x-label: none, - x-tick-step: none, x-ticks: ((3, $h_1$), (6, $h_2$)), x-grid: true, - y-label: none, y-tick-step: none, { - add(domain: (0, 3), t => calc.sin(t) + t) - add(domain: (3, 6), t => calc.sin(t) + t) - } - ) - }), $==>$, cetz.canvas({ - import cetz-plot.plot: * - plot( - size: (8, 4), axis-style: "school-book", x-label: none, - x-tick-step: none, x-ticks: ((6, $h_1$),), x-grid: true, y-label: none, - y-tick-step: none, { add(domain: (0, 6), t => calc.sin(t) + t) } +- Mixing discrete and continuous behaviours #pause +- Discrete steps are scheduled with _zero-crossing events_ + #grid(align: horizon + center, columns: (1fr, 4fr), { + let p(b) = cetz.canvas({ + cetz-plot.plot.plot( + axis-style: "left", size: (5, 3), + y-tick-step: none, y-label: none, + x-tick-step: none, x-label: none, b + ) + }) + let data = csv("data/ball.csv").map(r => r.map(float)) + alternatives-match(( + "1-3": p({ + cetz-plot.plot.add(data, line: "spline") + }), + "4-": p({ + cetz-plot.plot.add(data, line: "spline") + cetz-plot.plot.add(( + (3.2, 0), (8.3, 0), (12.4, 0), (15.6, 0), (18.2, 0), (20.3, 0), + (22.0, 0), (23.2, 0), (24, 0), (24.5, 0), (24.8, 0), (25, 0), + (25.1, 0), (25.2, 0), (25.3, 0), (25.4, 0), (25.5, 0), (25.6, 0), + (25.7, 0), (25.8, 0), (25.9, 0), (26.0, 0), (26.1, 0), (26.2, 0), + (26.3, 0), (26.4, 0), (26.5, 0), (26.6, 0), (26.7, 0), (26.8, 0), + (26.9, 0), (27.0, 0), (27.1, 0), (27.2, 0), (27.3, 0), (27.4, 0), + (27.5, 0), (27.6, 0), (27.7, 0), (27.8, 0), (27.9, 0), (28.0, 0), + ), mark: "o", mark-size: .1) + }) + )) + }, [ + #pause $ (d y)/(d t) = y' wide (d y')/(d t) = -g $ + #pause "When $y < 0$ at time $t$, $y'(t) = -k times (lim_(h -> t^-) y'(h))$" + ]) + #pause + ```zelus + let hybrid bouncing_ball(y0, y'0) = y where + rec der y = y' init y0 + and der y' = -g init y'0 reset up(-y) -> -k * (last y') + ``` + +#pagebreak() + +Compiled down to both a Mealy machine and an IVP \ #pause +Use of a zero-crossing detection mechanism@cit:illinois to monitor for events + +$ f_"zero" : Time times I times Y -> Z $ + +== Models with assertions + +// Observers rather than sub-nodes... +// Mention more clearly that they are executable +- In #lustre, assertions are observers on the parent state@cit:assertion_lustre +#only(2)[ +- Hybrid assertions are the same! + ```zelus + let hybrid sincos() = (sin, cos) where + rec der sin = cos init 0.0 + and der cos = -sin init 1.0 + and assert( + let der neg_cos = sin init -1.0 in + abs(neg_cos + cos) <= ε ) + ``` +] + +== Interferences + +Simultaneous simulation of independent systems causes interferences +#{ + let code1 = ```zelus + let hybrid ball() = p where + rec der p = v init 0.0 + and der v = a * (cos p) init 2.953 + and a = g * (sin p) - 5.0 * v / (cos p) + + let hybrid main() = p where + p = ball() + ``` + let code2 = ```zelus + let hybrid ball() = p where + rec der p = v init 0.0 + and der v = a * (cos p) init 2.953 + and a = g * (sin p) - 5.0 * v / (cos p) + + let hybrid main() = p where + p = ball() and _ = vdp(10.0) + ``` + let plot(file, x-step) = cetz.canvas({ + let x = csv(file).map( + row => (float(row.at(0).trim()), float(row.at(1).trim()))) + cetz-plot.plot.plot(axis-style: "left", size: (7, 2), y-tick-step: none, + y-max: calc.pi * 2, y-min: 0, x-grid: "major", + x-tick-step: none, x-label: none, y-label: none, legend: (6, 2), + x-ticks: ((x-step, ""),), { + cetz-plot.plot.add(x, label: `p`) + }) }) -) + let plot1(t) = plot("data/0000001.csv", t) + let plot2(t) = plot("data/10.csv", t) + let plot_ball(coords) = cetz.canvas({ + cetz-plot.plot.plot( + size: (7, 2), axis-style: "left", + x-label: `p`, y-label: [~~~], x-tick-step: none, y-tick-step: none, { + cetz-plot.plot.add(t => calc.cos(t), domain: (0, 13), samples: 500) + }) + cetz.draw.circle(coords, radius: .15) + }) + let anim1 = ( + ((0.2, 2.10), 00.0), ((1.0, 1.10), 00.5), ((1.8, 0.20), 01.0), + ((2.5, 1.30), 01.5), ((3.1, 2.10), 02.5), ((3.2, 2.15), 03.5), + ((3.1, 2.10), 04.5), ((2.5, 1.30), 06.0), ((1.8, 0.20), 07.5), + ((1.0, 1.10), 08.0), ((1.7, 0.15), 09.0), ((1.9, 0.30), 10.0), + ((1.7, 0.15), 12.0), ((1.5, 0.30), 14.0), ((1.7, 0.15), 18.0), + ((1.7, 0.15), 20.0), ((1.7, 0.15), 30.0), ((1.7, 0.15), 50.0), + ((1.7, 0.15), 60.0), + ).map(coords => grid(columns: (7fr, 3fr), align: center, code1, [ + #plot_ball(coords.at(0)) + #v(-10%) + #plot1(coords.at(1) / 2) + ])) + let anim2 = ( + ((0.2, 2.10), 00.0), ((1.0, 1.10), 00.5), ((1.8, 0.20), 01.0), + ((2.5, 1.30), 01.5), ((3.1, 2.10), 02.0), ((3.4, 2.15), 03.5), + ((3.7, 2.05), 05.5), ((4.3, 1.20), 07.0), ((5.1, 0.15), 07.5), + ((5.8, 1.15), 08.2), ((6.5, 2.05), 08.5), ((6.6, 2.15), 08.6), + ((6.5, 2.10), 09.2), ((5.9, 1.30), 09.8), ((5.2, 0.20), 10.2), + ((4.4, 1.10), 11.0), ((5.1, 0.15), 12.5), ((5.3, 0.30), 13.0), + ((5.1, 0.15), 14.0), ((4.9, 0.30), 16.0), ((5.1, 0.15), 20.0), + ((5.1, 0.15), 22.0), ((5.1, 0.15), 30.0), ((5.1, 0.15), 50.0), + ((5.1, 0.15), 60.0), + ).map(coords => grid(columns: (7fr, 3fr), align: center, code2, [ + #plot_ball(coords.at(0)) + #v(-10%) + #plot2(coords.at(1) / 2) + ])) + alternatives( + align(center, cetz.canvas({ + cetz-plot.plot.plot(size: (15, 5), x-tick-step: none, y-tick-step: none, + axis-style: "left", y-label: none, x-label: `p`, { + cetz-plot.plot.add(t => calc.cos(t), domain: (0, 13), samples: 500) + }) + cetz.draw.circle((0.4, 5.25), radius: 0.3) + cetz.draw.line((0.8, 5), (1.5, 4.5), mark: (end: "straight")) + })), + grid(columns: (7fr, 3fr), align: center, code1, []), ..anim1, + grid(columns: (7fr, 3fr), align: center, code2, []), ..anim2, + ) +} +#only(47)[This is the main issue with assertions] -= Composing simulations +== Summary -== Pushing values through +- Extension of a synchronous kernel to continuous time (```zelus der```, + ```zelus init```, ...) +- Scheduling of discrete steps using zero-crossing events (```zelus up```, ...) +- Execution using a numerical solver +- Simultaneous simulation causes interferences -#slide(self => [ - #let (uncover, only) = utils.methods(self) += Formalizing the Runtime - Unless we can copy the state, we need to use up all of a value before - producing the next. We need to push empty values ($bot$) as needed. +== Numerical Solvers - #align(center, cetz-canvas({ - import cetz.draw: * - box((0, 0), (3, 3), content: $M_1$) - box((8, 0), (3, 3), content: $M_2$) - only("1, 3-4", arrow((-1, 1.5), (r, 1))) - only("2, 5-", arrow((-1, 1.5), (r, 1), stroke: red)) - only("1, 2, 4-", arrow((3, 1.5), (r, 5))) - only("3", arrow((3, 1.5), (r, 5), stroke: red)) - only("-3", arrow((11, 1.5), (r, 1))) - only("4-", arrow((11, 1.5), (r, 1), stroke: red)) - content((-3, 1.5), $"Signal"(alpha)$) - content((5.5, 2), $"Signal"(beta)$) - content((14, 1.5), $"Signal"(gamma)$) +- Used to solve the model's initial value problem +- Integration is done _sequentially_ on successive intervals - content((-4, -1), $alpha:$) - content((-4, -2.5), $beta:$) - content((-4, -4), $gamma:$) - (pause,) - content((-2, -4), $bot$) - content((-2, -2.5), $bot$) - content((-2, -1.1), $alpha_1$) - (pause,) - content((0, -1), $bot$) - content((0, -2.6), $beta_1$) - content((0, -4), $bot$) - (pause,) - content((2, -1), $bot$) - content((2, -2.5), $bot$) - content((2, -4.1), $gamma_1$) - (pause,) - content((4, -1.1), $alpha_2$) - content((4, -2.5), $bot$) - content((4, -4.1), $gamma_2$) - content((5.5, -1), $...$) - content((5.5, -2.5), $...$) - content((5.5, -4), $...$) - (pause,) - box((3.5, -3), (1, 1), stroke: red) - box((0.1, 0.1), (2.8, 2.8), stroke: red) - content((12, -2.6), "Possible early reset!") - })) -]) +$ italic("csolve")(italic("ivp")) : + (h : Time) -> (h' : [0, h]) times (u : [0, h'] -> Y) $ -== Pulling values from +#pause -#slide(self => [ - #let (uncover, only) = utils.methods(self) +#align(center, { + import cetz-plot.plot: * + + let p(b, t) = cetz.canvas(plot( + size: (10, 2), axis-style: "school-book", legend: (9.2, 2.5), + y-tick-step: none, y-label: none, y-min: 0, y-max: 5, x-tick-step: none, + x-label: none, x-min: 0, x-max: 5.5, x-ticks: t, b + )) + alternatives( + p({ + add(domain: (0, 2), t => calc.sin(t) + t, label: $u$) + }, ((2, $h'$), (5, $h$))), + p({ + add(domain: (2, 4), t => calc.sin(t) + t, label: $u$) + add(domain: (0, 2), t => calc.sin(t) + t) + }, ((2, ""), (4, $h'$), (5, $h$))), + p({ + add(domain: (4, 5), t => calc.sin(t) + t, label: $u$) + add(domain: (0, 4), t => calc.sin(t) + t) + }, ((2, ""), (4, ""), (5, $h', h$))) + ) +}) #pause #pause #pause +#only("5-")[- Solvers maintain their own internal state] +#only("6-")[- Viewed as discrete nodes] - The output signal stops at least as often as the input signal does. - #linebreak() - This calls for a "lazy" composition: we request rather than provide data. +== Zero-crossing Detection - #align(center, cetz-canvas({ - import cetz.draw: * - box((0, 0), (3, 3), content: $M_1$) - box((8, 0), (3, 3), content: $M_2$) - only("1-3, 5-10, 12-", arrow((-1, 1.5), (r, 1))) - only("4, 11", arrow((-1, 1.5), (r, 1), stroke: red)) - only("1-2, 4-5, 7-8, 10-11, 13-", arrow((3, 1.5), (r, 5))) - only("3, 6, 9, 12", arrow((3, 1.5), (r, 5), stroke: red)) - only("1, 3-4, 6, 9, 11-12, 14-", arrow((11, 1.5), (r, 1))) - only("2, 5, 7, 8, 10, 13", arrow((11, 1.5), (r, 1), stroke: red)) - content((-3, 1.5), $"Signal"(alpha)$) - content((5.5, 2), $"Signal"(beta)$) - content((14, 1.5), $"Signal"(gamma)$) +- Detection of zero-crossing events using the integration results \ + $ italic("zsolve")(f_"zero") : (h : Time) times ([0, h] -> Y) -> (h' : [0, h]) times italic("Option")(Z_i) $ +#pause +#align(center, alternatives( + cetz.canvas({ + cetz-plot.plot.plot( + size: (10, 2), axis-style: "school-book", legend: (9, 2.5), + y-tick-step: none, y-label: none, y-min: -2, y-max: 5, + x-tick-step: none, x-label: none, x-min: 0, x-max: 2.2, + x-ticks: ((2/3, $h', h$),), { + cetz-plot.plot.add(domain: (0, 2/3), t => t * t - 1) + }) + }), + cetz.canvas({ + cetz-plot.plot.plot( + size: (10, 2), axis-style: "school-book", legend: (9, 2.5), + y-tick-step: none, y-label: none, y-min: -2, y-max: 5, + x-tick-step: none, x-label: none, x-min: 0, x-max: 2.2, + x-ticks: ((1, $h'$), (4/3, $h$)), { + cetz-plot.plot.add(domain: (2/3, 4/3), t => t * t - 1) + cetz-plot.plot.add(domain: (0, 2/3), t => t * t - 1) + }) + cetz.draw.line((4.55, 2.5), (4.55, 1.5), mark: (end: "straight")) + }), + cetz.canvas({ + cetz-plot.plot.plot( + size: (10, 2), axis-style: "school-book", legend: (9, 2.5), + y-tick-step: none, y-label: none, y-min: -2, y-max: 5, + x-tick-step: none, x-label: none, x-min: 0, x-max: 2.2, + x-ticks: ((1, ""), (2, $h', h$),), { + cetz-plot.plot.add(domain: (1, 2), t => -t * t + 3.5 * t) + cetz-plot.plot.add(domain: (0, 1), t => t * t - 1) + }) + }) +)) +#pause +- Discrete steps are performed when a zero-crossing event is detected #pause +- State changes may occur during discrete steps +#pause +- Viewed as discrete nodes - content((-4, -1), $alpha:$) - content((-4, -2.5), $beta:$) - content((-4, -4), $gamma:$) - (pause,) - content((-2, -4), $bot$) - (pause,) - content((-2, -2.5), $bot$) - (pause,) - content((-2, -1.1), $alpha_1$) - (pause,) - content((0, -4), $bot$) - (pause,) - content((0, -2.6), $beta_1$) - (pause,) - content((2, -4.1), $gamma_1$) - (pause,) - content((4, -4.1), $gamma_2$) - (pause,) - content((6, -2.6), $beta_2$) - content((6, -4), $bot$) - (pause,) - content((8, -4.1), $gamma_3$) - (pause,) - content((10, -4), $bot$) - content((10, -2.5), $bot$) - content((10, -1.1), $alpha_2$) - (pause,) - content((12, -2.6), $beta_3$) - content((12, -4), $bot$) - (pause,) - content((14, -4.1), $gamma_4$) - (pause,) - content((15.5, -1), $...$) - content((15.5, -2.5), $...$) - content((15.5, -4), $...$) - })) -]) +== Full simulation -= Conclusion +- Simulations are discrete nodes on streams of functions +- Two step modes: discrete and continuous -== What next? +#align(center + horizon, grid(columns: (1fr, 1fr), { + let a(s) = finite.automaton( + (D: (D: "", C: ""), + C: (C: "", D: "zero-crossing")), + initial: "D", final: (), layout: finite.layout.linear.with(spacing: 6), + style: s + ) + alternatives( + a((D: (fill: gray))), + a((C: (fill: gray), D-C: (stroke: red))), + a((C: (fill: gray), C-C: (stroke: red))), + a((D: (fill: gray), C-D: (stroke: red))), + a((C: (fill: gray), D-C: (stroke: red))), + a((C: (fill: gray), C-C: (stroke: red))), + a((D: (fill: gray), C-D: (stroke: red))), + a((D: (fill: gray), D-D: (stroke: red))), + a((C: (fill: gray), D-C: (stroke: red))), + ) +}, { + let p(b, t: ()) = cetz.canvas(cetz-plot.plot.plot( + size: (8, 5), axis-style: "school-book", + x-tick-step: none, x-label: none, x-min: 0, x-max: 10, + y-tick-step: none, y-label: none, y-min: -3, y-max: 10, b, + x-ticks: t, x-grid: "both" + )) + alternatives[#p({ + cetz-plot.plot.add(t => calc.sin(t - 1/(0.1*t+0.1)) + 8 + 0.1 * t, + domain: (0, 10), label: "in", samples: 500, style: (stroke: 2pt + blue)) + cetz-plot.plot.add(((0, 4),), label: "out", style: (stroke: 2pt + red)) + cetz-plot.plot.add(((0, -1),), label: "zero", style: (stroke: 2pt + green)) + })][#p({ + cetz-plot.plot.add(t => calc.sin(t - 1/(0.1*t+0.1)) + 8 + 0.1 * t, + domain: (0, 10), label: "in", samples: 500, style: (stroke: 2pt + blue)) + cetz-plot.plot.add(t => calc.cos(t) + 3, domain: (0, 2), label: "out", style: (stroke: 2pt + red)) + cetz-plot.plot.add(t => 0.2 * t * t - 1, domain: (0, 2), label: "zero", style: (stroke: 2pt + green)) + })][#p({ + cetz-plot.plot.add(t => calc.sin(t - 1/(0.1*t+0.1)) + 8 + 0.1 * t, + domain: (0, 10), label: "in", samples: 500, style: (stroke: 2pt + blue)) + cetz-plot.plot.add(t => calc.cos(t) + 3, domain: (0, 3), label: "out", style: (stroke: 2pt + red)) + cetz-plot.plot.add(t => 0.2 * t * t - 1, domain: (0, 3), label: "zero", style: (stroke: 2pt + green)) + }, t: ((2.25, ""),))][#p({ + cetz-plot.plot.add(t => calc.sin(t - 1/(0.1*t+0.1)) + 8 + 0.1 * t, + domain: (0, 10), label: "in", samples: 500, style: (stroke: 2pt + blue)) + cetz-plot.plot.add(t => calc.cos(t) + 3, domain: (0, 2.25), label: "out", style: (stroke: 2pt + red)) + cetz-plot.plot.add(t => 0.2 * t * t - 1, domain: (0, 2.25), label: "zero", style: (stroke: 2pt + green)) + cetz-plot.plot.add(((2.25, 3),), mark-style: (stroke: red + 2pt), mark: "+", mark-size: .04) + cetz-plot.plot.add(((2.25, -1),), mark-style: (stroke: green + 2pt), mark: "+", mark-size: .04) + })][#p({ + cetz-plot.plot.add(t => calc.sin(t - 1/(0.1*t+0.1)) + 8 + 0.1 * t, + domain: (0, 10), label: "in", samples: 500, style: (stroke: 2pt + blue)) + cetz-plot.plot.add(t => calc.cos(t) + 3, domain: (0, 2.25), label: "out", style: (stroke: 2pt + red)) + cetz-plot.plot.add(t => 0.2 * t * t - 1, domain: (0, 2.25), label: "zero", style: (stroke: 2pt + green)) + cetz-plot.plot.add(t => calc.cos(t) + 3.65, domain: (2.25, 6), style: (stroke: red + 2pt)) + cetz-plot.plot.add(t => calc.sin(t) + (0.2 * (t - 2.25)) - 1.8, domain: (2.25, 6), style: (stroke: green + 2pt)) + })][#p({ + cetz-plot.plot.add(t => calc.sin(t - 1/(0.1*t+0.1)) + 8 + 0.1 * t, + domain: (0, 10), label: "in", samples: 500, style: (stroke: 2pt + blue)) + cetz-plot.plot.add(t => calc.cos(t) + 3, domain: (0, 2.25), label: "out", style: (stroke: 2pt + red)) + cetz-plot.plot.add(t => 0.2 * t * t - 1, domain: (0, 2.25), label: "zero", style: (stroke: 2pt + green)) + cetz-plot.plot.add(t => calc.cos(t) + 3.65, domain: (2.25, 8), style: (stroke: red + 2pt)) + cetz-plot.plot.add(t => calc.sin(t) + (0.2 * (t - 2.25)) - 1.8, domain: (2.25, 8), style: (stroke: green + 2pt)) + }, t: ((7.2, ""),))][#p({ + cetz-plot.plot.add(t => calc.sin(t - 1/(0.1*t+0.1)) + 8 + 0.1 * t, + domain: (0, 10), label: "in", samples: 500, style: (stroke: 2pt + blue)) + cetz-plot.plot.add(t => calc.cos(t) + 3, domain: (0, 2.25), label: "out", style: (stroke: 2pt + red)) + cetz-plot.plot.add(t => 0.2 * t * t - 1, domain: (0, 2.25), label: "zero", style: (stroke: 2pt + green)) + cetz-plot.plot.add(t => calc.cos(t) + 3.65, domain: (2.25, 7.2), style: (stroke: red + 2pt)) + cetz-plot.plot.add(t => calc.sin(t) + (0.2 * (t - 2.25)) - 1.8, domain: (2.25, 7.2), style: (stroke: green + 2pt)) + cetz-plot.plot.add(((7.2, 5),), mark-style: (stroke: red + 2pt), mark: "+", mark-size: .04) + cetz-plot.plot.add(((7.2, 1),), mark-style: (stroke: green + 2pt), mark: "+", mark-size: .04) + })][#p({ + cetz-plot.plot.add(t => calc.sin(t - 1/(0.1*t+0.1)) + 8 + 0.1 * t, + domain: (0, 10), label: "in", samples: 500, style: (stroke: 2pt + blue)) + cetz-plot.plot.add(t => calc.cos(t) + 3, domain: (0, 2.25), label: "out", style: (stroke: 2pt + red)) + cetz-plot.plot.add(t => 0.2 * t * t - 1, domain: (0, 2.25), label: "zero", style: (stroke: 2pt + green)) + cetz-plot.plot.add(t => calc.cos(t) + 3.65, domain: (2.25, 7.2), style: (stroke: red + 2pt)) + cetz-plot.plot.add(t => calc.sin(t) + (0.2 * (t - 2.25)) - 1.8, domain: (2.25, 7.2), style: (stroke: green + 2pt)) + cetz-plot.plot.add(((7.2, 5),), mark-style: (stroke: red + 2pt), mark: "+", mark-size: .04) + cetz-plot.plot.add(((7.2, 1),), mark-style: (stroke: green + 2pt), mark: "+", mark-size: .04) + cetz-plot.plot.add(((7.2, 6),), mark-style: (stroke: red + 2pt), mark: "+", mark-size: .04) + cetz-plot.plot.add(((7.2, -1),), mark-style: (stroke: green + 2pt), mark: "+", mark-size: .04) + })][#p({ + cetz-plot.plot.add(t => calc.sin(t - 1/(0.1*t+0.1)) + 8 + 0.1 * t, + domain: (0, 10), label: "in", samples: 500, style: (stroke: 2pt + blue)) + cetz-plot.plot.add(t => calc.cos(t) + 3, domain: (0, 2.25), label: "out", style: (stroke: 2pt + red)) + cetz-plot.plot.add(t => 0.2 * t * t - 1, domain: (0, 2.25), label: "zero", style: (stroke: 2pt + green)) + cetz-plot.plot.add(t => calc.cos(t) + 3.65, domain: (2.25, 7.2), style: (stroke: red + 2pt)) + cetz-plot.plot.add(t => calc.sin(t) + (0.2 * (t - 2.25)) - 1.8, domain: (2.25, 7.2), style: (stroke: green + 2pt)) + cetz-plot.plot.add(((7.2, 5),), mark-style: (stroke: red + 2pt), mark: "+", mark-size: .04) + cetz-plot.plot.add(((7.2, 1),), mark-style: (stroke: green + 2pt), mark: "+", mark-size: .04) + cetz-plot.plot.add(t => t * (0.04 * calc.cos(t)) - 0.2 * (1 / t * t) + 6, domain: (7.2, 10), style: (stroke: red + 2pt)) + cetz-plot.plot.add(t => calc.sin(t) + (0.2 * (t - 2.25)) - 2.8, domain: (7.2, 10), style: (stroke: green + 2pt)) + })] +})) -#align(horizon, [ - - Can we avoid pointless solver resets? - - Think about assertions: is the checking of an assertion a composition? - - Plug to Zélus' output -]) +== Lifting simulations +- We can manipulate simulations directly in #zelus + +```zelus +type 'a dense = time * (time -> 'a) +type 'a signal = 'a dense option +val solve : ('a -C-> 'b) -S-> ('a signal -D-> 'b signal) +``` +#pause +- Running part of the model with its own solver instance is possible #pause +- This is exactly what we need for assertions! + +== Simulating assertions + +- Compile models with assertions down to two separate models +#pause +- Simulate the assertion independently with `solve` +#pause +- At each step of the parent model, run the assertion up to the horizon + +#align(center, { + let p(b, t) = cetz.canvas(cetz-plot.plot.plot( + size: (10, 4), axis-style: "left", + x-label: none, y-label: none, x-tick-step: none, y-tick-step: none, + y-min: 0, x-min: 0.1, x-max: 10, x-ticks: t, + b + )) + import cetz-plot.plot: add + alternatives( + p({ + add(t => calc.sin(t) + 5, domain: (0, 5), label: "Parent", samples: 500) + add(((0, 0),), label: "Assertion", samples: 500) + }, ((5, ""),)), + p({ + add(t => calc.sin(t) + 5, domain: (0, 5), label: "Parent", samples: 500) + add(t => calc.cos(2 * t) + 2, domain: (0, 2), label: "Assertion", samples: 500) + }, ((5, ""), (2, ""))), + p({ + add(t => calc.sin(t) + 5, domain: (0, 5), label: "Parent", samples: 500) + add(t => calc.cos(2 * t) + 2, domain: (0, 5), label: "Assertion", samples: 500) + }, ((5, ""), (2, ""))), + p({ + add(t => calc.sin(t) + 5, domain: (0, 10), label: "Parent", samples: 500) + add(t => calc.cos(2 * t) + 2, domain: (0, 5), label: "Assertion", samples: 500) + }, ((5, ""), (2, ""), (10, ""))), + p({ + add(t => calc.sin(t) + 5, domain: (0, 10), label: "Parent", samples: 500) + add(t => calc.cos(2 * t) + 2, domain: (0, 7), label: "Assertion", samples: 500) + }, ((5, ""), (2, ""), (10, ""), (7, ""))), + p({ + add(t => calc.sin(t) + 5, domain: (0, 10), label: "Parent", samples: 500) + add(t => calc.cos(2 * t) + 2, domain: (0, 9), label: "Assertion", samples: 500) + }, ((5, ""), (2, ""), (10, ""), (7, ""), (9, ""))), + p({ + add(t => calc.sin(t) + 5, domain: (0, 10), label: "Parent", samples: 500) + add(t => calc.cos(2 * t) + 2, domain: (0, 10), label: "Assertion", samples: 500) + }, ((5, ""), (2, ""), (10, ""), (7, ""), (9, ""))), + ) +}) + += Conclusion + +== Conclusion + +*Results* +- Efficient implementation of a hybrid system simulation runtime +- Primitives to allow manipulation of simulations directly in #zelus +- Assertions representable as continuous observers +#pause + +*Future Work* +- Implementation of a compilation pass (in progress) +- Support for _nested assertions_ +- Concerns related to lifting simulations (can we use ```zelus pre```)? diff --git a/doc/sources.bib b/doc/sources.bib index 6c32c75..75bfca5 100644 --- a/doc/sources.bib +++ b/doc/sources.bib @@ -300,3 +300,22 @@ pages = {265–292}, language = {en}, } +@article{cit:mealy, + title = {A Method for Synthesizing Sequential Circuits}, + volume = {34}, + ISSN = {1538-7305}, + DOI = {10.1002/j.1538-7305.1955.tb03788.x}, + abstractNote = {The theoretical basis of sequential circuit synthesis is + developed, with particular reference to the work of D. A. + Huffman and E. F. Moore. A new method of synthesis is + developed which emphasizes formal procedures rather than the + more familiar intuitive ones. Familiarity is assumed with the + use of switching algebra in the synthesis of combinational + circuits.}, + number = {5}, + journal = {Bell System Technical Journal}, + author = {Mealy, George H.}, + year = {1955}, + pages = {1045–1079}, + language = {en}, +} diff --git a/doc/zelus.sublime-syntax b/doc/zelus.sublime-syntax index 44022a5..67e9c7d 100644 --- a/doc/zelus.sublime-syntax +++ b/doc/zelus.sublime-syntax @@ -6,7 +6,7 @@ scope: source contexts: main: - - match: \b(val|let|in|where|rec|and|local)\b + - match: \b(type|val|let|in|where|rec|and|local)\b scope: keyword.control - match: \b(if|then|else|match|with)\b @@ -18,7 +18,7 @@ contexts: - match: \b(period|up|assert|der|init|reset|present|last|fby|pre)\b scope: keyword.control - - match: \b(float|int)\b + - match: \b(float|int|bool)\b scope: entity.name.type - match: \b[0-9]*(\.[0-9]+)?\b