Controllability properties of dynamical M. Zoppello systems with - - PowerPoint PPT Presentation

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Controllability properties of dynamical M. Zoppello systems with - - PowerPoint PPT Presentation

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Controllability properties of dynamical systems with hysteresis Controllability properties of dynamical M. Zoppello systems with hysteresis Motivations Hysteresis Properties of the Play operator Marta Zoppello joint work with F. Bagagiolo

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SLIDE 1

Controllability properties of dynamical systems with hysteresis

  • M. Zoppello

Motivations Hysteresis

Properties of the Play

  • perator

Hysteresis applied

  • n the controls

Hysteresis acting

  • n the state

An example Idea

Idea for generalization

Perspectives and

  • pen problems

Controllability properties of dynamical systems with hysteresis

Marta Zoppello joint work with F. Bagagiolo

“Hysteresis and controllability of affine driftless systems” Submitted

Control of state constrained dynamical systems Padova, September 25 - 29, 2017

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SLIDE 2

Controllability properties of dynamical systems with hysteresis

  • M. Zoppello

Motivations Hysteresis

Properties of the Play

  • perator

Hysteresis applied

  • n the controls

Hysteresis acting

  • n the state

An example Idea

Idea for generalization

Perspectives and

  • pen problems

Motivations

Many dynamical systems present delay phenomena:

◮ Gear systems, ◮ Hydraulic controlled

valves,

◮ Systems governed by a

magnetic field for example magnetic micro-swimmers ⇒ MEMORY EFFECT

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SLIDE 3

Controllability properties of dynamical systems with hysteresis

  • M. Zoppello

Motivations Hysteresis

Properties of the Play

  • perator

Hysteresis applied

  • n the controls

Hysteresis acting

  • n the state

An example Idea

Idea for generalization

Perspectives and

  • pen problems

Hysteresis

One way of representing mathematically this memory effect is the use the so called hysteresis operators

ρ" #ρ" 0" u" w"

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SLIDE 4

Controllability properties of dynamical systems with hysteresis

  • M. Zoppello

Motivations Hysteresis

Properties of the Play

  • perator

Hysteresis applied

  • n the controls

Hysteresis acting

  • n the state

An example Idea

Idea for generalization

Perspectives and

  • pen problems

Properties of the Play operator

F : C0([0, T]) × B → C0([0, T])

a) Causality: u|[0,t] = v|[0,t] ⇒ F[u, w0](t) = F[v, w0](t) b) Rate independence: F[u ◦ φ, w0] = F[u, w0] ◦ φ ∀φ continuous non decreasing c) Lipschitz continuity: ||F[u, w1

0 ] − F[v, w2 0 ]||C0([0,T]) ≤ L(||u − v||C0([0,T]) + ||w1 0 − w2 0 ||B)

d) semigroup property: F[u, w0](t) = F[u|[τ,t], F[u, w0](τ)](t − τ)

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SLIDE 5

Controllability properties of dynamical systems with hysteresis

  • M. Zoppello

Motivations Hysteresis

Properties of the Play

  • perator

Hysteresis applied

  • n the controls

Hysteresis acting

  • n the state

An example Idea

Idea for generalization

Perspectives and

  • pen problems

˙ z =

m

  • i=1

gi(z)ui Hysteresis Play operator ւ ց On the controls On the state ˙ z = m

i=1 gi(z)F[ui, w0]

˙ z = m

i=1 gi(F[z, w0])ui

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SLIDE 6

Controllability properties of dynamical systems with hysteresis

  • M. Zoppello

Motivations Hysteresis

Properties of the Play

  • perator

Hysteresis applied

  • n the controls

Hysteresis acting

  • n the state

An example Idea

Idea for generalization

Perspectives and

  • pen problems

Hysteresis applied on the controls

˙ z =

m

  • i=1

gi(z)ui (1) ˙ z =

m

  • i=1

gi(z)F[vi, u0] (2)

Theorem 1: Approximating sequence Let us suppose that the system (1) is controllable in time T and let ¯ u be the piecewise constant con- trol which steers the system between two fixed configurations in time T, then we are always able to find a sequence of continuous functions vk = (vik)m

i=1, such that uk = F[vk, ¯

u0] converges to ¯ u in L1([0, T]).

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SLIDE 7

Controllability properties of dynamical systems with hysteresis

  • M. Zoppello

Motivations Hysteresis

Properties of the Play

  • perator

Hysteresis applied

  • n the controls

Hysteresis acting

  • n the state

An example Idea

Idea for generalization

Perspectives and

  • pen problems

◮ Find a sequence uk ∈ C0

1 2 3 4 1.0 0.5 0.5 1.0 1.5 2.0

s.t. lim

k→∞ uk i = ¯

ui in L1([0, T]) ∀i = 1 · · · m

1 2 3 4 t 1.0 0.5 0.5 1.0 1.5 2.0 uk 1 2 3 4 t 1.0 0.5 0.5 1.0 1.5 2.0 vk

Lemma

The play operator is surjective on the set of the ziggurat functions

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SLIDE 8

Controllability properties of dynamical systems with hysteresis

  • M. Zoppello

Motivations Hysteresis

Properties of the Play

  • perator

Hysteresis applied

  • n the controls

Hysteresis acting

  • n the state

An example Idea

Idea for generalization

Perspectives and

  • pen problems

Theorem 2: The trajectory of the system (2) with the controls vk defined in Theorem 1 converges to the trajectory of the non hysteretic system (1) with controls ¯ u.

Proof.

||zk − z||∞ ≤ Ck + mMLt||zk − z||∞ where Ck = mt||gi||∞ t

0 |F[vi k(s)] − ¯

ui(s)| ds → 0 for the convergence of ui

k to ¯

ui in L1. The last inequality for the Gronwal lemma implies that ||zk − z||∞ ≤ CkemMLt → 0

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SLIDE 9

Controllability properties of dynamical systems with hysteresis

  • M. Zoppello

Motivations Hysteresis

Properties of the Play

  • perator

Hysteresis applied

  • n the controls

Hysteresis acting

  • n the state

An example Idea

Idea for generalization

Perspectives and

  • pen problems

Hysteresis acting on the state ˙ z =

m

  • i=1

gi(F[z, w0])ui (3)

Questions

◮ In which cases we are able to obtain controllability

results?

◮ Classical Lie algebra conditions are still applicable? ◮ Which kind of techniques are applicable?

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SLIDE 10

Controllability properties of dynamical systems with hysteresis

  • M. Zoppello

Motivations Hysteresis

Properties of the Play

  • perator

Hysteresis applied

  • n the controls

Hysteresis acting

  • n the state

An example Idea

Idea for generalization

Perspectives and

  • pen problems

An example

Figure: The system of the Heisenberg flywheel

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SLIDE 11

Controllability properties of dynamical systems with hysteresis

  • M. Zoppello

Motivations Hysteresis

Properties of the Play

  • perator

Hysteresis applied

  • n the controls

Hysteresis acting

  • n the state

An example Idea

Idea for generalization

Perspectives and

  • pen problems

Consider the following control system   ˙ x ˙ y ˙ z   =   1   u1 +   1 f(x)   u2 (4) and its hysteretic version   ˙ x ˙ y ˙ z   =   1   u1 +   1 f(F[x, w0])   u2 (5)

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SLIDE 12

Controllability properties of dynamical systems with hysteresis

  • M. Zoppello

Motivations Hysteresis

Properties of the Play

  • perator

Hysteresis applied

  • n the controls

Hysteresis acting

  • n the state

An example Idea

Idea for generalization

Perspectives and

  • pen problems

Idea

  • 1. Approximate the linear trajectory generated by u1

Theorem 3: The play operator has dense image in the space of piecewise linear continuous functions

1 2 3 4 t 0.5 1.0 1.5 2.0 2.5 x 1 2 3 4 t 0.5 1.0 1.5 2.0 2.5 vj

  • 2. Use it to reach the final yB and zB
  • 3. Adjust the last coordinate
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SLIDE 13

Controllability properties of dynamical systems with hysteresis

  • M. Zoppello

Motivations Hysteresis

Properties of the Play

  • perator

Hysteresis applied

  • n the controls

Hysteresis acting

  • n the state

An example Idea

Idea for generalization

Perspectives and

  • pen problems

Proposition

Given ¯ x(t) the piecewise linear continuous trajectory of the non hysteretic system we are able to find a sequence vj(t) such that F[vj, w0](t) → ¯ x(t) in L∞ as j → ∞.

Theorem

For any initial and final configurations A and B and for any suitable w0, there always exists a sequence of piecewise constant controls (uj

1, uj 2) and a final time T ∗ such that

the solution (xj(t), yj(t), zj(t)) of system (5) starting from A is such that xj(T ∗) = xB yj(T ∗) = yB zj(T ∗) → zB as j → ∞

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SLIDE 14

Controllability properties of dynamical systems with hysteresis

  • M. Zoppello

Motivations Hysteresis

Properties of the Play

  • perator

Hysteresis applied

  • n the controls

Hysteresis acting

  • n the state

An example Idea

Idea for generalization

Perspectives and

  • pen problems

Idea for generalization

The same result can be reached iterating the procedure if

  • ne has the following kind of system

z = (x1 · · · xm, ym+1, · · · y2m−1) g1(z) = ∂x1 g2(z) = ∂x2 + f2(F[x1, w0])∂ym+1 . . . gm(z) = ∂xm + fm(F[x1, w0], · · · , F[xm−1, w0])∂y2m−1 (6)

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SLIDE 15

Controllability properties of dynamical systems with hysteresis

  • M. Zoppello

Motivations Hysteresis

Properties of the Play

  • perator

Hysteresis applied

  • n the controls

Hysteresis acting

  • n the state

An example Idea

Idea for generalization

Perspectives and

  • pen problems

Sketch of the proof

◮ Adjust the last coordinate y2m−1 using an appropriate

sequence of controls

◮ Put um ≡ 0 and use another sequence of controls

uj

1, · · · , uj m−1 to adjust the coordinate y2m−2. ◮ Repeat the procedure

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SLIDE 16

Controllability properties of dynamical systems with hysteresis

  • M. Zoppello

Motivations Hysteresis

Properties of the Play

  • perator

Hysteresis applied

  • n the controls

Hysteresis acting

  • n the state

An example Idea

Idea for generalization

Perspectives and

  • pen problems

Perspectives and open problems

◮ Are the last kind of systems wide enough? Are they

all Carnot groups?

◮ Is it possible to have some “Chow-like” theorem? ◮ Use other hysteresis operators ◮ this techniques are applicable to the magnetic

micro-swimmer system?

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SLIDE 17

Controllability properties of dynamical systems with hysteresis

  • M. Zoppello

Motivations Hysteresis

Properties of the Play

  • perator

Hysteresis applied

  • n the controls

Hysteresis acting

  • n the state

An example Idea

Idea for generalization

Perspectives and

  • pen problems

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