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

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 Hysteresis applied “Hysteresis and controllability of affine driftless systems” Submitted on the controls Hysteresis acting on the state An example Idea Idea for generalization Perspectives and open problems Control of state constrained dynamical systems Padova, September 25 - 29, 2017

Controllability Motivations properties of dynamical systems with hysteresis M. Zoppello Motivations Many dynamical systems present delay phenomena: Hysteresis Properties of the Play operator Hysteresis applied ◮ Gear systems, on the controls Hysteresis acting ◮ Hydraulic controlled on the state valves, An example ⇒ MEMORY EFFECT ◮ Systems governed by a Idea Idea for generalization magnetic field for Perspectives and open problems example magnetic micro-swimmers

Controllability Hysteresis properties of dynamical systems One way of representing mathematically this memory effect is the use with hysteresis the so called hysteresis operators M. Zoppello Motivations Hysteresis w" Properties of the Play operator Hysteresis applied on the controls #ρ" 0" ρ" u" Hysteresis acting on the state An example Idea Idea for generalization Perspectives and open problems

Controllability Properties of the Play operator properties of dynamical systems with hysteresis M. Zoppello F : C 0 ([ 0 , T ]) × B → C 0 ([ 0 , T ]) Motivations Hysteresis Properties of the Play operator Hysteresis applied on the controls Hysteresis acting on the state An example Idea Idea for generalization Perspectives and a) Causality: u | [ 0 , t ] = v | [ 0 , t ] ⇒ F [ u , w 0 ]( t ) = F [ v , w 0 ]( t ) open problems b) Rate independence: F [ u ◦ φ, w 0 ] = F [ u , w 0 ] ◦ φ ∀ φ continuous non decreasing c) Lipschitz continuity: ||F [ u , w 1 0 ] − F [ v , w 2 0 ] || C 0 ([ 0 , T ]) ≤ L ( || u − v || C 0 ([ 0 , T ]) + || w 1 0 − w 2 0 || B ) d) semigroup property: F [ u , w 0 ]( t ) = F [ u | [ τ, t ] , F [ u , w 0 ]( τ )]( t − τ )

Controllability properties of dynamical systems with hysteresis M. Zoppello m Motivations � ˙ z = g i ( z ) u i Hysteresis Properties of the Play i = 1 operator Hysteresis applied on the controls Hysteresis Play operator Hysteresis acting on the state ւ ց An example Idea Idea for generalization On the controls On the state Perspectives and open problems z = � m z = � m ˙ ˙ i = 1 g i ( z ) F [ u i , w 0 ] i = 1 g i ( F [ z , w 0 ]) u i

Controllability Hysteresis applied on the controls properties of dynamical systems with hysteresis M. Zoppello m m � ˙ z = g i ( z ) u i (1) � ˙ z = g i ( z ) F [ v i , u 0 ] (2) Motivations i = 1 i = 1 Hysteresis Properties of the Play operator Hysteresis applied on the controls Theorem 1: Approximating sequence Hysteresis acting on the state An example Let us suppose that the system (1) is controllable Idea in time T and let ¯ u be the piecewise constant con- Idea for generalization trol which steers the system between two fixed Perspectives and open problems configurations in time T , then we are always able to find a sequence of continuous functions v k = ( v i k ) m i = 1 , such that u k = F [ v k , ¯ u 0 ] converges to ¯ u in L 1 ([ 0 , T ]) .

Controllability ◮ Find a sequence u k ∈ C 0 properties of dynamical systems 2.0 with hysteresis 1.5 M. Zoppello 1.0 0.5 Motivations 1 2 3 4 Hysteresis � 0.5 Properties of the Play operator � 1.0 Hysteresis applied s.t. on the controls k →∞ u k in L 1 ([ 0 , T ]) ∀ i = 1 · · · m i = ¯ lim u i Hysteresis acting on the state An example u k 2.0 Idea 1.5 Idea for generalization 1.0 Perspectives and 0.5 Lemma open problems t 1 2 3 4 � 0.5 The play operator is � 1.0 v k surjective on the set of 2.0 the ziggurat 1.5 1.0 functions 0.5 t 1 2 3 4 � 0.5 � 1.0

Controllability properties of Theorem 2: dynamical systems with hysteresis The trajectory of the system (2) with the controls v k M. Zoppello defined in Theorem 1 converges to the trajectory of Motivations the non hysteretic system (1) with controls ¯ u . Hysteresis Properties of the Play operator Hysteresis applied Proof. on the controls Hysteresis acting on the state || z k − z || ∞ ≤ C k + mMLt || z k − z || ∞ An example � t Idea 0 |F [ v i u i ( s ) | ds → 0 for the k ( s )] − ¯ where C k = mt || g i || ∞ Idea for generalization u i in L 1 . The last inequality for the convergence of u i k to ¯ Perspectives and open problems Gronwal lemma implies that || z k − z || ∞ ≤ C k e mMLt → 0

Controllability Hysteresis acting on the state properties of dynamical systems with hysteresis M. Zoppello m Motivations � ˙ z = g i ( F [ z , w 0 ]) u i (3) Hysteresis Properties of the Play operator i = 1 Hysteresis applied on the controls Hysteresis acting Questions on the state An example Idea ◮ In which cases we are able to obtain controllability Idea for generalization results? Perspectives and open problems ◮ Classical Lie algebra conditions are still applicable? ◮ Which kind of techniques are applicable?

Controllability An example properties of dynamical systems with hysteresis M. Zoppello Motivations Hysteresis Properties of the Play operator Hysteresis applied on the controls Hysteresis acting on the state An example Idea Idea for generalization Perspectives and open problems Figure: The system of the Heisenberg flywheel

Controllability properties of dynamical systems with hysteresis M. Zoppello Consider the following control system Motivations ˙  x   1   0  Hysteresis Properties of the Play  =  u 1 +  u 2 ˙ operator y 0 1 (4)    Hysteresis applied ˙ f ( x ) z 0 on the controls Hysteresis acting on the state and its hysteretic version An example  ˙      x 1 0 Idea Idea for generalization  =  u 1 +  u 2 ˙ y 0 1 (5)    Perspectives and ˙ open problems z 0 f ( F [ x , w 0 ])

Controllability Idea properties of dynamical systems with hysteresis M. Zoppello 1. Approximate the linear trajectory generated by u 1 Theorem 3: Motivations Hysteresis The play operator has dense image in the Properties of the Play operator space of piecewise linear continuous functions Hysteresis applied on the controls Hysteresis acting on the state x v j 2.5 An example 2.5 2.0 2.0 Idea Idea for generalization 1.5 1.5 Perspectives and 1.0 1.0 open problems 0.5 0.5 t t 1 2 3 4 1 2 3 4 2. Use it to reach the final y B and z B 3. Adjust the last coordinate

Controllability properties of dynamical systems with hysteresis Proposition M. Zoppello Given ¯ x ( t ) the piecewise linear continuous trajectory of Motivations the non hysteretic system we are able to find a sequence Hysteresis x ( t ) in L ∞ as j → ∞ . v j ( t ) such that F [ v j , w 0 ]( t ) → ¯ Properties of the Play operator Hysteresis applied on the controls Theorem Hysteresis acting on the state For any initial and final configurations A and B and for any An example suitable w 0 , there always exists a sequence of piecewise Idea 2 ) and a final time T ∗ such that constant controls ( u j 1 , u j Idea for generalization the solution ( x j ( t ) , y j ( t ) , z j ( t )) of system (5) starting from Perspectives and open problems A is such that x j ( T ∗ ) = x B y j ( T ∗ ) = y B z j ( T ∗ ) → z B as j → ∞

Controllability Idea for generalization properties of dynamical systems with hysteresis M. Zoppello Motivations The same result can be reached iterating the procedure if Hysteresis Properties of the Play one has the following kind of system operator z = ( x 1 · · · x m , y m + 1 , · · · y 2 m − 1 ) Hysteresis applied on the controls Hysteresis acting g 1 ( z ) = ∂ x 1 on the state An example g 2 ( z ) = ∂ x 2 + f 2 ( F [ x 1 , w 0 ]) ∂ y m + 1 Idea (6) . . Idea for generalization . Perspectives and open problems g m ( z ) = ∂ x m + f m ( F [ x 1 , w 0 ] , · · · , F [ x m − 1 , w 0 ]) ∂ y 2 m − 1

Controllability Sketch of the proof properties of dynamical systems with hysteresis M. Zoppello Motivations Hysteresis Properties of the Play operator ◮ Adjust the last coordinate y 2 m − 1 using an appropriate Hysteresis applied on the controls sequence of controls Hysteresis acting ◮ Put u m ≡ 0 and use another sequence of controls on the state u j 1 , · · · , u j An example m − 1 to adjust the coordinate y 2 m − 2 . Idea ◮ Repeat the procedure Idea for generalization Perspectives and open problems

Controllability Perspectives and open problems properties of dynamical systems with hysteresis M. Zoppello Motivations Hysteresis Properties of the Play operator ◮ Are the last kind of systems wide enough? Are they Hysteresis applied all Carnot groups? on the controls Hysteresis acting ◮ Is it possible to have some “Chow-like” theorem? on the state ◮ Use other hysteresis operators An example Idea ◮ this techniques are applicable to the magnetic Idea for generalization micro-swimmer system? Perspectives and open problems