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 PWI 1 Mapping: x n +1 = f d ( x n ) = R θ x n + 2H d ( − u n sin θ + v n cos θ ) ( − cot θ, 1) T where x n = ( u n , v n ) T , R θ is a clockwise rotation by θ and H d ( x ) = − k ∈ ❩ such that 2 k − 1 ≤ x < 2 k + 1 . Figure: Action of f d on its invariant rhombus for θ = π/ 3. 1 A.C. Davies, Phil Trans: Phys Sci & Eng, vol. 353 no. 1701 (1995)

Engineering example II: Σ − ∆ modulator PWI 2 Mapping: x n +1 = f s ( x n ) = R θ x n + H s ( x n ) ( − cot θ, 1) T where x n = ( u n , v n ) T and H s ( x n ) = sgn ( u n sin θ + v n cos θ ) − 2 cos θ sgn v n . Figure: The action of f s on its invariant set, M , with θ = 1 . 8 . 2 P. Ashwin, JHBD, X-C. Fu, Proc ISCAS 2001 pp III-811 – III-814 (2001)

Engineering example III: Buck converter 3 ◮ After appropriate linear transformations, buck converter dynamics are described by � q ( z − C 0 ) + C 0 Re z < 0 z ′ = T q ( z ) = q ( z − C 1 ) + C 1 Re z > 0 with C 0 , C 1 ∈ ❈ and exactly one of C 0 , C 1 = − 1. ◮ Choose C 0 = − 1 here: choice has non-trivial consequences. ◮ Parameter q = re i θ ∈ ❈ . ◮ Realistic modelling forces | q | < 1. ◮ Goetz map has | q | = 1 and is a limiting case of the contracting map. 3 JHBD, 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 PW C s 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 T q ? 4 Henk Bruin & JHBD, Proc AMS vol. 137 1389–1395 (2009)

Example of PWC dynamics Basins of attraction for T q with q = re i θ = 0 . 99 × exp 0 . 75i, C 0 = 0 . 36 − 0 . 5i, C 1 = − 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 = [ ν 1 , ν 2 . . . , ν p ] with ν i ∈ { 0 , 1 } — by simulation. ◮ This requires large searches: ∀ q ∈ ❈ , | q | < 1, and ∀ C 1 ∈ ❈ where C 1 = second centre of rotation. In practice, computational effort goes into solving sets of p linear, simultaneous inequalities to establish existence or not of C 1 for given p , ν : ◮ Fix p & ν = [ ν 1 , ν 2 . . . , ν p ]. ◮ From T q , compute period- p orbit, z 0 , . . . , z p − 1 , with z p = z 0 . ◮ Check that, for i = 0 . . . p − 1, � 0 Re z i < 0 ν i = 1 Re z i > 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 orbit 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 orbit 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 ) = z 0 , z 1 . . . , z p − 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). ∼ 2 p − 1 / p of these; c.f. 2 p 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 C 1 , y = Im C 1 for S ( p , ν ) to exist. In practice ◮ it is feasible to carry out a search over ∼ 10 6 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: � � � � c 1 r + O ( r 2 ) − s 1 r + O ( r 2 ) x + y − 1 + O ( r ) < 0 � � � � c 2 r 2 + O ( r 3 ) − s 2 r 2 + O ( r 3 ) y − 1 + O ( r 2 ) x + < 0 � � � � c 3 r 3 + O ( r 4 ) − s 3 r 3 + O ( r 4 ) y − 1 + O ( r 3 ) x + < 0 � c 4 r 4 + O ( r 5 ) � � − s 4 r 4 + O ( r 5 ) � y − 1 + O ( r 4 ) + < 0 x � c 5 r 5 + O ( r 6 ) � � − s 5 r 5 + O ( r 6 ) � y +1 + O ( r 5 ) − x − < 0 � � s 1 r + O ( r 6 ) − c 1 r + O ( r 6 ) < 0 [1 + O ( r )] x + y where q = r e i θ , c i = cos i θ, s i = sin i θ ; C 1 = x + i y . ◮ 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. C 0 = − 1, C 1 ∈ ❈ .

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 2 6 = 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 p − 1 � �� � Fix p ∈ ◆ and coding ν = [1 , 0 . . . 0] . Then, for all p > 1 , ∃ θ ∈ (0 , π ) such that a periodic orbit of the mapping T q with this coding exists as | q | → 0 . Proof The p inequalities to be satisfied, to lowest order in r , are − c p − 1 r p − 1 x + s p − 1 r p − 1 y + 1 < 0 x + s 1 ry − c 1 r < 0; and c k r k x − s k r k y − 1 < 0 , for k = 1 . . . p − 2. Substituting y = 0, assuming x < 0 and choosing θ such that c 1 . . . c p − 2 > 0 but c p − 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 π 2( p − 1) , satisfies the required constraints on 2( p − 2) c 1 . . . c p − 1 . � � ◮ Numerics strongly indicate that in fact | θ | ∈ π 0 , for p − 2 p > 3. ◮ The real part of C 1 , 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 of 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?