Piecewise Isometries and Piecewise Contractions in Electronic - - PowerPoint PPT Presentation

Piecewise Isometries and Piecewise Contractions in Electronic Engineering

Jonathan Deane Department of Mathematics University of Surrey Guildford, Surrey, UK e-mail: J.Deane@surrey.ac.uk

Overview

◮ Three PWI/PWC electronic engineering examples:

• 1. Second order digital filter
• 2. Σ − ∆ modulator
• 3. Buck converter

◮ The case for piecewise contractions. ◮ PWCs are asymptotically periodic: what next? ◮ Global dynamics of the contracting Goetz map:

• 1. Useful lemmas
• 2. Simulations: ‘eclipse diagrams’
• 3. The strong dissipation limit.

◮ Conclusions & further work.

Engineering example I: digital filter PWI1

Mapping: xn+1 = fd(xn) = Rθxn + 2Hd(−un sin θ + vn cos θ) (− cot θ, 1)T where xn = (un, vn)T, Rθ is a clockwise rotation by θ and Hd(x) = −k ∈ ❩ such that 2k − 1 ≤ x < 2k + 1.

Figure: Action of fd on its invariant rhombus for θ = π/3.

1A.C. Davies, Phil Trans: Phys Sci & Eng, vol. 353 no. 1701 (1995)

Engineering example II: Σ − ∆ modulator PWI2

Mapping: xn+1 = fs(xn) = Rθxn + Hs(xn) (− cot θ, 1)T where xn = (un, vn)T and Hs(xn) = sgn(un sin θ + vn cos θ) − 2 cos θ sgn vn.

Figure: The action of fs on its invariant set, M, with θ = 1.8.

• 2P. Ashwin, JHBD, X-C. Fu, Proc ISCAS 2001 pp III-811 – III-814 (2001)

Engineering example III: Buck converter3

◮ After appropriate linear transformations, buck converter

dynamics are described by z′ = Tq(z) =

• q(z − C0) + C0

Re z < 0 q(z − C1) + C1 Re z > 0 with C0, C1 ∈ ❈ and exactly one of C0, C1 = −1.

◮ Choose C0 = −1 here: choice has non-trivial consequences. ◮ Parameter q = reiθ ∈ ❈. ◮ Realistic modelling forces |q| < 1. ◮ Goetz map has |q| = 1 and is a limiting case of the

contracting map.

3JHBD, Proc NOLTA 2004, pp 147 – 150 (2004)

Dissipation: PWIs vs. PWCs

◮ In all three examples, neglecting dissipation is unrealistic. ◮ Dissipation forces contraction. ◮ Hence, good motivation for studying PWCs in connection

with electronic engineering applications.

◮ General result proved for all planar PWCs:

Piecewise contractions are asymptotically periodic.4

◮ So PWC dynamics are trivial then?

If not. . . What can we say about the relationship between q and the (periodic) dynamics of Tq?

4Henk Bruin & JHBD, Proc AMS vol. 137 1389–1395 (2009)

Example of PWC dynamics

Basins of attraction for Tq with q = reiθ = 0.99 × exp 0.75i, C0 = 0.36 − 0.5i, C1 = −1, Re z ∈ [−1, 1], Im z ∈ [0, 2]. Periods 2 (blue), 4 (black), 5 (yellow), 11 (red) and 30 (green).

PWC dynamics — continued

◮ “Periodic” is not necessarily “simple”. ◮ Seemingly arbitrary periods and coexisting attractors

complicate things.

◮ However, for strong dissipation — small r — we might expect

simple dynamics. Raises questions; e.g., as r → 0, I Upper bound to the period of any periodic solutions? II Upper bound to the number of co-existing solutions?

Numerical investigation

◮ Investigate various periods p & codings ν = [ν1, ν2 . . . , νp]

with νi ∈ {0, 1} — by simulation.

◮ This requires large searches: ∀q ∈ ❈, |q| < 1, and ∀C1 ∈ ❈

where C1 = second centre of rotation. In practice, computational effort goes into solving sets of p linear, simultaneous inequalities to establish existence or not of C1 for given p, ν:

◮ Fix p & ν = [ν1, ν2 . . . , νp]. ◮ From Tq, compute period-p orbit, z0, . . . , zp−1, with zp = z0. ◮ Check that, for i = 0 . . . p − 1,

νi =

• Re zi < 0

1 Re zi > 0.

Useful lemmas

• 0. If q ∈ ❘, p = 1.
• 1. The coding of every period-p orbit with p > 1 must contain

at least one 0 and at least one 1.

• 2. If a period-p > 1 orbit with a coding ν exists, then a period-p
• rbit with a coding [ν(i+j) mod p, i = 0 . . . p − 1], for all j, also

exists. [1000 = ⇒ 0001, 0010, 0100]

• 3. If a period-p > 1 orbit with a coding ν exists, then a period-p
• rbit with a coding ν′ also exists, where ν′ = ν with 0 ↔ 1.

[1000 = ⇒ 0111]

• 4. If a period-p orbit with coding ν exists for a given q, then

such an orbit also exists for q∗.

• 5. Let p have a non-trivial factorisation, so that p = nm. Then

there exist no non-degenerate period-p orbits whose codings consist of n identical blocks of length m. [No 1010 for p = 4.]

Minimising the search

Let S(p, ν) = z0, z1 . . . , zp−1 be a period-p orbit with coding ν. Need to consider

◮ small period p ◮ ν = a subset of necklace sequences only (Lemmas 1–5

above). ∼ 2p−1/p of these; c.f. 2p length-p binary sequences. Also need

◮ efficient algorithm (e.g. Fourier-Motzkin; Intersection; L.P.

simplex method; Farkas’ Lemma) to find a solution, if one exists, to the p linear inequalities in x = Re C1, y = Im C1 for S(p, ν) to exist. In practice

◮ it is feasible to carry out a search over ∼ 106 q-values in unit

disc, for p ≤ 20.

Example: inequalities for period-6, ν = [1, 0, 0, 0, 0, 0]

Find x, y ∈ ❘ such that the following are satisfied:

• c1r + O(r2)
• x

+

• −s1r + O(r2)
• y

−1 + O(r) < 0

• c2r2 + O(r3)
• x

+

• −s2r2 + O(r3)
• y −1 + O(r2)

< 0

• c3r3 + O(r4)
• x

+

• −s3r3 + O(r4)
• y −1 + O(r3)

< 0

• c4r4 + O(r5)
• x

+

• −s4r4 + O(r5)
• y −1 + O(r4)

< 0 −

• c5r5 + O(r6)
• x −
• −s5r5 + O(r6)
• y +1 + O(r5)

< 0 [1 + O(r)] x +

• s1r + O(r6)
• y

−c1r + O(r6) < 0 where q = reiθ, ci = cos iθ, si = sin iθ; C1 = x + iy.

◮ Scaling problems in numerics for small r.

Results: ‘eclipse’ diagrams for all period-6 solutions

Shows the unit disc |q| < 1; black: solution exists; white: no solution exists with this coding. C0 = −1, C1 ∈ ❈.

Results: what ‘eclipse’ diagrams tell us

• 1. Dichotomy between cases in which a solution exists

∀r ∈ (0, 1], for some θ; and those in which this is not true.

• 2. No solution exists for all θ and small r.
• 3. Note the q → q∗ symmetry (Lemma 4).
• 4. Real axis excluded by Lemma 0.
• 5. Only 5 out of the possible 26 = 64 possible solutions had to

be investigated (Lemmas 1, 2, 3, & 5). 1–4 appear to be true for all p ≤ 20 investigated. No solutions with p > 1 can exist for r = 0 (Lemma 0).

Results: numbers of codings for which solutions exist

Results: fraction of unit disc where a period-p solution exists

Answering question I

Theorem

Fix p ∈ ◆ and coding ν = [1,

p−1

0 . . . 0]. Then, for all p > 1, ∃θ ∈ (0, π) such that a periodic orbit of the mapping Tq with this coding exists as |q| → 0. Proof The p inequalities to be satisfied, to lowest order in r, are x + s1ry − c1r < 0; −cp−1rp−1x + sp−1rp−1y + 1 < 0 and ckrkx − skrky − 1 < 0, for k = 1 . . . p − 2. Substituting y = 0, assuming x < 0 and choosing θ such that c1 . . . cp−2 > 0 but cp−1 < 0 clearly satisfies all the inequalities, for arbitrarily small positive r — provided only that x is sufficiently negative. ✷

Remarks on the theorem

◮ |θ| ∈

• π

2(p−1), π 2(p−2)

• satisfies the required constraints on

c1 . . . cp−1.

◮ Numerics strongly indicate that in fact |θ| ∈

• 0,

π p−2

• for

p > 3.

◮ The real part of C1, i.e. x, scales as r−p+1

— second centre moves away from the origin as r → 0.

◮ Question I is therefore answered: there is no upper limit to

the period of solutions, even as r → 0.

Conclusions

◮ Dissipative system =

⇒ contracting mapping.

◮ Motivates study of piecewise contractions (PWCs) in

engineering systems.

◮ Dynamics of PWCs, despite always being eventually periodic,

non-trivial.

◮ Attempt to build up a preliminary picture of global behaviour

• f PWCs through simulations of the dynamics.

◮ Sketch of proof that solutions of arbitrarily long period exist,

even for strong contraction.

Further work

For arbitrarily small r (a.s.r.):

◮ Analyse other codings: find other families of solutions that

exist for a.s.r.

◮ Work on the co-existence problem and answer Question II. ◮ What can we prove about solutions that do not exist for any θ

as r → 0? For all r ∈ (0, 1):

◮ Other features of eclipse diagrams (e.g. θ increasing with r;

single region ⊂ (0, π) for a.s.r. solutions).

◮ What makes a solution with a given coding exist or not?