sistemas hybridos

8/14/2019 sistemas hybridos

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Dept of Electrical and Electronic Eng.Dept of Electrical and Electronic Eng.

PhD School on Electronic and Information Eng.

Zeno phenomena in hybrid systems

and

sliding mode behaviours

Elio USAI

[email protected]


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SummarySummary

Hybrid Systems

Zeno phenomena Mechanical systems

Optimal control

Sliding Modes Invariance

Equivalent control

Filippovs solution

Approximability

Advances

Zeno phenomena in hybrid systems and sliding mode behaviours


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Hybrid SystemsHybrid Systems

Dynamical systems

Possible different dynamics

Switchings between dynamics

Autonomous

Forced

Jumps in the state

Autonomous (often due to constraints)

Forced



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Various dynamics


( ) ( ) ( )( )

N

RRR

= +

Qi

tUXtttft m

i

n

ii ,,,,, uxuxx&

fiis a smooth vector field fi:RnxRmxR+xRn

The state dynamics is invariant unti l a switch occurs

Jumps in the state may not cause changes of the dynamics


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Jumps in the state variables


( ) ( ) ( )( )

( ) ( ) ( )( )

{ }

N

RRR

=

=

+

+

Qi

tUX

r

g

lm

i

n

i

i

kkkk

jm

kkkk

i

1,0,,

,,

0,,,

vux

uxx

vux

The reaching of the guardgijm causes the jump, according

to proper rules, from the state x(k) to a new state which is

the new initial state for the system dynamics


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Hybrid system dynamics may behave very differently from

each of the constituting ones.


( ) ( ) ( )

( ) ( ) ( )2010

201

101

0

20

10

aa

systemtyatyaty

systemtyatyaty

0 the systems are both asymptotically stable a1=0 the systems are both marginally stable

a1


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-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

y(t)

dy(t)/dt

Phase plane

a0

1

=0.5

a0

2

=9.0

-10 -5 0 5 10-5

0

5

10

15

20Phase plane

y(t)

dy(

t)/dt

( ) ( )

( ) ( )2010

020

10

aasystemtyty

systemtyty


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( ) ( )

( ) ( )2010

020

10

aasystemtyty

systemtyty


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Q N: set of discrete statesD Rn : set of domains of continuous statesE Q xQ : set of edges defining relations between domains

G : set of functions defining the guardsR : set of reset functions

U Rm : set of domains of continuous controlsS {0,1}l : set of domains of discrete controls

{ }SUFRGEDQ ,,,,,,,=H

Not all sets have to appear in the definition of a hybrid system


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T= {i}iN : set of switsching/jump time instantsIn = {xi} iNxiD : set of initial states sequenceEd = {i}iNi=(i,j)Q xQ : set of edge sequence

{ }EdInT ,,=H

The execution H of the hybrid system H can be constituted byfinite or infinite elements.

Executions with infinite number of elements may be due to theZenoZenophenomenon


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Zeno phenomenaZeno phenomena


The Zeno phenomenon appears when the execution

H of the hybrid system is such that

( )


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In a Zeno condition the system evolves along a guard

( )( ) ( ) = ttttg 0, x

x

x&

The guard is an invariant set of the system state

After the Zeno time the system is constrained to

evolve on the guard

Such a constrain may be natural or forced


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The bouncing ball( )

( ) ( )( ) ( ) [ ]1,0,2,10:

0

===

=

+

iiyy

iy

ygty

ii

&&

K

&&

y

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [s]

Elev

ation( )

+++=

+==

+=

+

++

1

12

2

2

0

210

1

0

2

0

221

0

0

0

gyyy

gyyyy

gyy

gg

k

kk

k

gkk

&&

&&&

&


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The bouncing ball

( )

( ) ( )( ) ( ) [ ]1,0

,2,10:

00

===

+=

+

iiyy

iy

ygty

ii

&&

K

&&

Representation as a linearcomplementarity system

Representation as a differential

inclusion( ) [ ) ,gty&&

Such a differential inclusion representsmore than a bouncing ball


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Mass-spring and Coulomb friction

( ) ( )( ) ( ) 0sgn =++ tkytybtym &&&

Switching time instantsm

k

y( )

k

mky kk == 0&

-6 -4 -2 0 2 4 6-5

-4

-3

-2

-1

0

1

2

3

4

5Phase space

y(t)

dy(t)/dt

0 2 4 6 8 10-4

-3

-2

-1

0

1

2

3

4

5

Time [s]

velocityposition


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Singular optimal control

( ) ( )

( ) ( )[ ]

( ) ( ) T

T

Tu

dtubfJ

ubf

xxxx

xx

xxx

==

+=

+=

,0,1 0

0

00

&

( ) ( ) ( ) ( )[ ]

( ) ( )( )xpxx

p

xxpxx

bbu

H

ubfubfH

T

T

T

+=

=

+++=

0

00

sgn

&

( ) ( ) [ ]210 ,0 tttbb T =+ xpx u switches at infinite frequency ?


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Continuation of the trajectory

Regularization Averaging

Filippov solution

Equivalent control

Physical meaning of the Zeno behaviour

Modeling errors Limit behaviour


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SlidingSliding modesmodes


Special case of Zeno behaviour in switching systems

The system is forced onto a surface in the statespace, i.e., the sliding surface

The system is invariant when constrained on the

sliding surface

When the system is constrained on the sliding

surface, the system modes differ from those of the

original systems

Any system on the sliding surface behaves the same

way


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Sliding modesSliding modes


( )

,2.0,1,1

0sgn

21

21

===

+==+

caa

cyyyayay

&

&&&

-0.5 0 0.5 1 1.5

-0.25

-0.2

-0.15

-0.1

-0.05

0

y

dy/dt

The state slides along the

sliding manifold


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Sliding modesSliding modes -- InvarianceInvariance


( ) ( )

( ) ( ) ( ) ( )tbbFtf

utbtfxnixx

m

n

ii

,0,,

,,1,2,11

xxxx

xx


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=

=

+

+=

+===

1

1

1

1

1

1 2,2,1

n

i

iin

n

i

iin

ii

xcx

xcx

nixx

&

K&

The system is invariant when constrained on the sliding manifold

The system behaves as a

reduced order system with

prescribed eigenvalues

Matching uncertainties, included in the uncertain functionf, are

completely rejected

In the sliding mode it is not possible to recover the originalsystem dynamics (semi-group property)

Take care of unstable zero-dynamics


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( ) ( ) ( ) ( ) ( )cyy

tuyykkybyybbytm

+=

=+++++

&

&&&&&

sinsgn 231321

( )sgnUu =

0 0.5 1 1.5 2 2.5 3 3.5-5

-4

-3

-2

-1

0

1

2

3

4

5

Time [s]

positionvelocity


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Sliding modesSliding modes EquivalentEquivalent controlcontrol


The equivalent control is the control signal that assures

0&

( ) ( )

( )

( ) 0,

,

,,

1

1

1

1

1

1

1

1

+

=

++=

+=

=

+

=+

=

&

&

tb

xctf

u

xcutbtf

xcx

n

i

ii

eq

n

i

ii

n

i

iin

x

x

xx

If the sliding mode behaviour is detected, the control variable

could be switched to the equivalent control


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The equivalent control is not available, in practice, because of uncertainty

( ) ( ) ( )=

+=

jUjUjU eq

The equivalent control can be estimated by a low-pass filter

uuu avav =+If ueq is bounded with its time derivative then

=

eqav uu0

0lim


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The cut-off frequency of the low-pass filter must be

Greater than the bandwidth of the equivalent control

Lower than the real switching frequency

In practice only an estimate of ueq can be evaluated

Finding the equivalent control can be a continuation

method in Zeno hybrid systems


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The bouncing ball

( )

( ) ( )( ) ( ) [ ]1,0

,2,10:

00

=

==

+=

+

iiyy

iy

ygty

ii

&&

K

&& ( )

0

0

=

=

=

y

y

tgt

&

Mass-spring and Coulomb friction

( ) ( )( ) ( ) 0sgn =++ tkytybtym &&&

( )( ) ( )

[ ]kb

kb

bk

y

y

tyty

+=

=

,

0

sgn

&

&


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SlidingSliding modesmodes Equivalent controlEquivalent control


( ) ( ) ( ) ( ) ( )cyy

tuyykkybyybbytm

+=

=+++++

&

&&&&&

sinsgn 231321

( ) ( ) ( ) ( ) ( )ytcmtyykkybyybbueq &&&& +++++++= sinsgn 231321

0 0.5 1 1.5 2 2.5 3 3.5-30

-25

-20

-15

-10

-5

0

5

10

15

Time [s]

green: ideal ueq

blue: estimateduav


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( ) ( )[ ]UUtbtf ++= ,,, xxx F&

The controlled system dynamics belongs to a differential inclusion

The sliding variable can be considered as a performance

index to be nullified to find the right solution

( ) ( )F+=

equtbtf ,,* xxx&


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Sliding modesSliding modes Filippov solutionFilippov solution


The average velocity is defined by

If the system dynamics is affine in the control variable theequivalent control and the Filippov solution agree

( )( )( )

( )( )( )

,1,+

+

+

=

=

grad

grad

grad

grad

If the system dynamics is nonlinear the Filippov solutionmay be not unique


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Sliding modesSliding modesApproximabilityApproximability


In the case of switching errors the switching frequency isfsthe real trajectory x(t) is near to the ideal one x*(t) and

( ) ( ) 0* sf

tt xx

( ) ( )[ ]( ) ( ) ( ) xxxx

xxxNMtutbtf

UUtbtf++

++=,,,

,,,&

( )

=

0x

xxC


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By the Bellman-Gronwall lemma

( )

( ) ( ) ( ) ( ) +

++

HdtttLStt

dtNMTMt

T

T

0

**

0

0

xxxx

xxx

D=+

0

&

If is the equivalent delay of the switching


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The most common cause of delay switching is the digital

implementation of the controller

0 1 2 3 4 5 6-8

-6

-4

-2

0

2

4

6

8x 10

-3

Tempo [s]

=1e-4

( ) ( ) ( )cyy tuyykkyybbytm += =++++ &&&&

sin

2

3121

0 1 2 3 4 5 6-8

-6

-4

-2

0

2

4

6

8x 10

-4

Time [s]

=1e-5


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The approximability property allows for obtaining significant

simulations of the sliding mode behaviour, even in the presence

of a Zeno phenomenon

Anyway, care should be given

Fixed step integration methods should be used

Analysis of local numerical instability may be necessary


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Sliding modesSliding modesAdvancesAdvances


The control algorithms for Higher-Order Sliding Modes require

differentiators

Relativedegree Feedback signals VSS controller

1 Classical 1-SMC, Super-Twisting( )ssign

2

( ) ( )ssignssign &, Twisting 2-SMC

( ){ }ssigns &, Sub-optimal 2-SMC

3

ss &,

Any

rUniversal HOSM)(,,...,, )1()2( rr ssignsss &

( ) ( ) ( )ssignssignssign &&& ,, Hybrid 3-VSC

Dynamic / Terminal 2-SMC


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Sliding modesSliding modes --AdvancesAdvances


x1

k2

k3

m 2

b 3 F

m 3 x3

x 2

m c(t)

kc(t) b c(t)

k1

b 2

b 1

x c

c

( ) ( ) ( ) ( )( ) ( )tty

tttt

Cz

BfzAz

=

+=&

[ ]Txxxxxx332211

,,,,, &&&=z

( )

( )( )

( )( ) ( ) ( )

=

++

++

++

3

32

3

32

3

2

3

2

2

2

2

2

2

21

2

21

2

1

2

1

1111

00

100000

001000

00

000010

m

bb

m

kk

m

b

m

k

m

b

m

k

m

bb

m

kk

m

b

m

k

tm

b

tm

k

tm

btb

tm

ktk

ccc

c

c

c

tA

[ ]00

0

00

0

00

00

00

1111

1

1

3

2bkbk

m

m =

= CB


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SlidingSliding modesmodes --AdvancesAdvances


The relative degree between the control forcefc and the output

(i.e., the contact force ) is r= 1.

By means of an cascade integrator the control force results tobe continuous and the relative degree becomesr = 2

If the Sub-Optimal second-order sliding mode controller is

implemented

( )exUu sgn =

The contact force is regulated to the reference value in

spite of uncertainties on the parameter variations of the

suspended catenary systems, and on the pantograph itself.

[ )( ) ( ) 0tsuch thalasttheis

1;0

=

exexex tt

&


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Sliding modesSliding modes --AdvancesAdvances


0 50 100 150 200 250 300 350 40090

95

100

105

110

Contactforce

0 50 100 150 200 250 300 350 400-100

-50

0

50

Train speed [km/h]

upperframe

controlforce


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ReferencesReferences


V.I. Utkin, Variable Structure Systems with Sliding Modes, IEEE Transactions on Automatic Control, vol.22,

no.2 , pp. 212-222, April 1977

G. Bartolini, T. Zolezzi, Variable Structure Systems Nonlinear in the Control Law, IEEE Transactions on

Automatic Control, vol.30, no.3 , pp. 681-684, July 1985

A. Levant, Slidng order and sliding accuracy in sliding mode control, International Journal of Control, vol. 58,no. 6, 1247-1263, 1993

S. Drakunov, Sliding Modes in Hybrid Systems A Semigroup Approach, IEEE-CDC 1994, 4235-4240, Lake

Buena Vista, Florida, USA, December 1994

K.H. Johansson, J. Lygerros, S. Sastry, M. Egerstedt, Simulation of Zeno Hybrid Automata, IEEE-CDC 1999,

3538-3543, Phenix, Arizona, USA, December 1999

M.K. Calibel, J.M. Schumacher, On the Zeno behavior of linear complementarity systems, IEEE-CDC 2001,

346-351, Orlando, Florida, USA, December 2001

Heymann M, Feng Lin, Meyer G, Resmerita S., Analysis of Zeno behaviors in a class of hybrid systems, IEEE

Transactions on Automatic Control, vol.50, no.3 , pp. 376-83, March 2005.

A.D. Ames, S. Sastry, Characterization of Zeno Behavior in Hybrid Systems Using Homological Methods,

2005 American Control Conference, pp. 1160-1165, Portland, Oregon, USA, June 2005

A. Abate, A.D. Ames, S. Sastry, Stochastic Approximations of Hybrid Systems, 2005 American Control

Conference, pp. 1557-1562, Portland, Oregon, USA, June 2005

A.D. Ames , Z. Haiyang , R.D. Gregg , S. Sastry Is there life after Zeno? Taking executions past the breaking

(Zeno) point. 2006 American Control Conference, pp. 2652-2657, Minneapolis, Minnesota, USA, June 2006


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ReferencesReferences


V.I. Utkin, Sliding Modes in Control Optimization, Springer-Verlag, Berlin, 1992

A.F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers,

Dordrecht, 1988

M.I. Zelikin, V.F. Borisov, Theory of Chattering Control, with applications to Astronautics, Robotics, Economics,

and Engineering, Birkhuser, Boston, 1994

C. Edwards, E. Fossas, L. Fridman eds., Advances in Variable Structure and Sliding Mode Control, LNCIS 334,

Springer-Verlag, Berlin, 2006

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