Hybrid Dynamical Systems Talk

8/12/2019 Hybrid Dynamical Systems Talk

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Hybrid Dynamical Systems

H. A. Blair1 D. W. Jakel1 R. J. Irwin1 A. Rivera2

1Syracuse University{blair,dwjakel,rjirwin}@ecs.syr.edu

2Utica [email protected]

January 24, 2008

Revised May 15, 2008

H. A. Blair, D. W. Jakel, R. J. Irwin, A. Rivera (Syracuse University)

Hybrid Systems 01/03/08 1 / 35
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What & Why

What:Discrete state transition rules and differential equations

symmetrically combined.Why:There is a notion of formal hybrid dynamical system which

has a semantics that allows for discrete, continuous and hybrid

models of these systems.




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Plan of the talk

1 Brief summary of results

2

Examples of hybrid programs3 Continuity and convergence structure

4 Differentials

5 Conclusions


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Brief summary of results

Differentiation satisfying the chain-rule in the Cartesian-closed

category of convergence spaces.All topological spaces and directed graphs, together with hybrid

amalgamations are convergence spaces.

Some of the resulting differential calculi conservatively extend

elementary differential calculus on Euclidean and Hilbert spaces.

Some convergence spaces admit regular actions

(example: translations on a Euclidean space).

These include topological vector spaces and modules over rings

and lead to a unique maximum differential calculus that

generalizes linear functions as differentials.Models of first-order logic expand to convergence spaces. Permits

a differential operator on formulas.

ForX,Ytopological spaces,Xlocally compact: The space of

continuous functions fromX toY, has the compact open topology

as its convergence structure. (D.W. Jakel,2007).H. A. Blair, D. W. Jakel, R. J. Irwin, A. Rivera (Syracuse University)Hybrid Systems 01/03/08 4 / 35


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Example 1

For this example, view the atoms of ground logic program Pas

Truth-valued functions of integers (i.e. discrete sequential

time-steps).

The program clauses constitute deterministic constraints on these

functions via the one-step consequence operatorTP.

H. A. Blair, D. W. Jakel, R. J. Irwin, A. Rivera (Syracuse University)Hybrid Systems 01/03/08 6 / 35
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Example 1, continued

e.g.p q, pr q

View the above two clauses as

p(t+ 1) = q(t), p(t)r(t+ 1) = q(t)q(t+ 1) = false

Generally, the program must first be transformed to K. Clarks(Clark, 78) definitional form: at most one clause abouteach

predicate.

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Example 1, continued

Temporarily ad hoc: (well get this as a theorem a little later), for

each atomA, let

Adenote the change ofAper time step,

where

A(t) =T indicates A(t+ 1) =A(t).

A(t) =F indicates A(t+ 1) =A(t).

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Example 1, Differential equations form

The program in this example transforms to

p = (q p) p

q = false q

r = q r

Together with the initial conditions (i.e. the values ofp,qandrat0) these equations determine

t=0 t=1 t=2

p F F F

q F F F

r F T T

p F F F

q F F F

r T F F

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Example 2: Continuous time

Let atomAbe a function from the real numbers Rto the Booleanvalues B.

We take for the set of differentials both functions

f(t) = b ift=0

F ift=0

whereb B. This way, we get two differentials: forb=F andb=T.

Let

A(t0)denote the differential ofAatt0. We will obtain thefollowing as a theorem: Ais differentiable at t0 iffA(t0+ t) A(t0)is identical to one of the two differentials on some sufficiently small

punctured neighborhood of 0.



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Example 2: Continuous time

Derivatives aredifferential-valued.

Identify the differentials with the Boolean values F and T,

respectively.

Then the following makes sense:

p = (q p) p

r = q r

q = false q

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Going hybrid

Combining types:

p = (q p) p

y = f(p, r, y)

r = q r

q = false q

s = g(y, p) s

How do we make principled sense of all this?

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Continuity - basic idea

No jumps in the output

Question:How do we make principled sense of this?

Answer:We pick out a structure that functions to detect jumps.

A continuous line is one that can be drawn without lifting the pencil

from the paper.

Dana Scott writes:

In other words, no jumps, no lifting the chalk (pen/pencil). As youdraw, you have to stay close to previous points. Is there anything

deeper here? Does the definition not make an intuitive idea

rigorous?

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A definition by ostension (pointing out)

Consider the flow of time from time t0 to timet1.

Whether or not this is real, we have an intuition that time flows

continuously.

Represent that intuition by a straight line segment.

t1t0

Definition:Whatever continuity is, this line segment is continuous.

Anything else that bears the right relationships - yet to be given -to this line segment is also continuous.

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Some more examples

continuous

not continuous

What relationship does the continuous curve bear to our

continuous straight line segment that the discontinuous curve doesnt bear?

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"The right relationship"

a b

P(a)

P(b)

Q(a) Q(b)

Time

Path P

Path Q

t

P(t)

Q(t)

On path P, think of P(a) as the position of a trav-eling point at time a. Etc. On each path, there

is an interval between the two red points.

What are we left with on the Time line and on

the paths as the interval demarcated by the red

points on the Time line shrinks down onto theblack point?

On the continuous path labeled Time, we get

just the one point at time t. Similarly on path Q

we get just the one black point Q(t). But on path

P we get two points, P(t) together with the point

at the left boundary of the upper segment. This

is our jump detector.

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Neighborhood systems

Not every kind of space contains intervals and lines. We want to

understand continuity in such spaces. Without intervals, what can

we do?

Answer:Neighborhood Systems

A neighborhood system at a point p in a space is a collection ofregions of the space, each such region containing p - call them

neighborhoods ofp- such that the overlap of any two of the

neighborhoods ofpis also a neighborhood ofpandevery region

of the space containing a neighborhood of pis itself a

neighborhood ofp.

Another term for a neighborhood system is filter.

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Filters - Example



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Convergence structure

Filteron a set X: A nonempty collection of subsets ofXclosed

under (i) pairwise intersection and (ii) reverse inclusion.

Convergence space: A setXtogether with a convergence

structure; i.e. a relation (we writeFxand say thatF convergesto x) from the set of filters onX toXsatisfying two constraints:1 For all pointsxand filtersF andG: ifFxandGF, thenGx.2 {A| xA} x. We call{A| xA}apointfilter, and denote it by

[x].

[Beautiful paper including a brief excellent tutorial on convergencespaces: R. Heckmann TCS v.305, (159186)(2003)]

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Topology as convergence structure

Suppose:at each pointxthere is a smallest filter converging tox -

theneighborhood filter. Then the convergence structure is said to

bepretopological.

A neighborhood ofxis a member of the neighborhood filter

converging tox.Bourbaki, 1948. A set isopenif, and only if, it is a neighborhood

of each of its member points.

Definition:A convergence structure onX istopologicalif, and only

if, it is pretopological and there is some topology on Xsuch that ateach pointxthe neighborhood filter converging toxcontains all

and only the supersets of the open sets containing x.

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Directed graphs as Convergence spaces

Thepre-topological representation:

Pretopological convergence spaceXrepresents a directed graph

if, for each pointx, the neighborhood filter converging toxhas a

smallest member.

The edge relation for the directed graph is: There is an edge from

x toy iffyN(x), whereN(x)is the smallest set in the smallestfilter converging tox. (Specialization relation: yis contained in

every neighborhood ofx.)

Proposition:The convergence structure of pre-topological

representation of a digraph is topological if, and only if, the edgerelation is transitive. (Specialization preorder.)

There are other ways to represent directed graphs by

convergence spaces:

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Post-discrete convergence spaces

Definition:A convergence space ispost-discreteif every proper

filter (not containing the empty set) converging somewhere is a

point filter.

Every post-discrete convergence space represents a directed

graph and conversely: There is an edge from x toy iff[y]x.

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Post-discrete convergence spaces:Example

The point filters converging to 0 are[0]and[1].

The only point filter converging to 1 is [1].

The pre-topological representation of this digraph is aka Sierpinski

space - call this theSierpinski digraph.

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Continuity

Definition:(Kent, 1954)f : (X, X)(Y, Y)iscontinuousatx iffmaps every filter converging toxto a collection of images that

generate a filter converging tof(x).

Let CONV be the Cartesian closed category whose objects are

convergence spaces, and arrows are continuous functions.The category of topological spaces is embedded in CONV. The

category of reflexive digraphs is embedded also. Continuity for

reflexive digraphs is homomorphism; i.e. edge preservation.

More detail on the embeddings can be found inBlair, Irwin, Jakel, Rivera, 2007, Proceedings of the Symposium on

Logical Foundations of Computer Science, CUNY, June, 2007, Springer.

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Continuity

Definition:(Kent, 1954)f : (X, X)(Y, Y)iscontinuousatx iffmaps every filter converging toxto a collection of images that

generate a filter converging tof(x).

On digraphs, continuity is equivalent to edge-preservation with

respect to both pretopological representation and post-discreterepresentation; i.e. graph-homomorphism.

Key idea: Convergence structure on a space Ycan be set up so

that distinctly different views of what is salient in it appear by

looking at it with continuous functions coming in from other

spaces. Different convergence structures in other spaces cangenerally only see part of the convergence structure in the space

they are looking at.

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A message

A space can appear as a discrete structure or as a continuous

structure, depending who or what is doing the looking!

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Continuity:Example

f : RSierp

Pre-topologically: f is continuous iff fis the test function of anopen subset of R.

Post-discretely: f is continuous ifff is constant.


X
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Exponentiating; i.e. YX

(Heckmann 03, Hyland 79, Binz 66)Letf YX, letFbe a filter onYX. ThenF f iff for everyxXand filterF x: F(F) f(x).

This is the largest convergence structure (thats good, the filtersare small, like having a small topology) such that f.f(x)iscontinuous onYX X.

Also, consequently: composition is continuous.

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Local continuity

filterG Y f(x) filter F

X x

G-neighborhoodi.e. a G-neighborhood is just

a member of G

VF-neighborhoodU

xU :

Input Output

f(x) V


L l Diff i bili
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Local Differentiability

filter FX x

G-neighborhoodi.e. a G-neighborhood is just

a member of G

VF-neighborhood U

xU :

Input Output

h G: h(x) =f(x)

filterG DIFF(X,Y)g


Ch i R l
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Chain Rule

Theorem:

Supposedf Diff(X, Y),dgDiff(Y, Z)and d(gf) Diff(X, Z).

Supposedf is a differential off atx0.Supposedgis a differential of gatf(x0).

Then,d(gf)is a differential ofgf atx0.


S l ti d l i t t
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Selecting and applying convergence structures - our

examples

Begin with the ordinary Euclidean topology on R.

Make a copy

Rof R, but with punctured neighborhoods.

f :

R Ris continuous if, and only if, f : R Ris continuous

But,

Rallows for discrete change at xwhen mapping into a space

equipped with a hybrid convergence structure.

For real-valued functions, begin with the Euclidean topology and

then allow every point filter converging anywhere to converge

everywhere. Again, does not change the continuous functions.This move superposes a post discrete complete graph structure

on the reals to allow for continuous embeddings of discrete

structures.


S l ti g d l i g g t t
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needs

In gauge theory in physics, for example, it is necessary to map

discrete structure into continuous structure. For the reals with its

usual topology, one is confined to functions that are constant on

connected components. Blcch!

The reals have to look discrete with respect to the external view

from a discrete structure. We must hybridize the reals to get this.

But not so, going back from the reals to pretopological discrete

structures - which gives rise to interesting combinatorial puzzlesabout coloring fractally arranged intervals.


Selecting and applying convergence structures our
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examples

Rserves as continuous time.

Post-discrete representation ofK2 serves as the Boolean values.

Post discrete representation of the reflexive closure of the

successor relation on the integers serves as discrete time.


Take home message
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Take-home message

We have a means to craft seamless hybriddynamical systems

with a precisely given semantics.

We have a full hybrid differential calculusto aid in reasoningabout such systems.

(Angel Rivera, 08) A logic with convergence space models for

reasoning about properties of flows of hybrid dynamical systems.

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Hybrid Dynamical Systems Talk