Delayed feedback of sampled higher derivatives

Tamas Insperger€, Gabor Stepan€, Janos Turi$

€Department of Applied MechanicsBudapest University of Technology and Economics

$Programs in Mathematical SciencesUniversity of Texas at Dallas

Contents- Stability gained with time-periodic parameters

- Human balancing (delay and threshold)

- The labyrinth and the eye – a mechanical view

- Robotic balancing (sampling and round-off)

- Micro-chaos (stable & unstable)

- Segway – without gyros

- Retarded, neutral and advanced FDEs (linear)

- Stability achieved with sampled higher derivatives

- Conclusions

The delayed Mathieu equation

Analytically constructed stability chart for testing numerical methods and algorithms

Time delay and time periodicity are equal:

Mathieu equation (1868)

Delayed oscillator (1941)

)2()()cos()( txbtxttx

2T0b0

Stability chart – Mathieu equation

Floquet (1883)

Hill (1886)

Rayleigh(1887)

van der Pol &

Strutt (1928)

Strutt – Ince diagram (1956)

0)()cos()( txttx

Stephenson (1908), Swinney (2004), Zelei (2005)

Swing (2000BC)

Stability chart – delayed oscillator

Vyshnegradskii… Pontryagin (1942) Nyquist (1949) Bellman & Cooke (1963)

Hsu & Bhatt (1966) Olgac (2000)

)2()()( txbtxtx

The delayed Mathieu – stability charts

b=0

ε=1 ε=0

)2()()cos()( txbtxttx

Stability chart of delayed Mathieu

Insperger, Stepan Proc Roy Soc A (2002)

)2()()cos()( txbtxttx

Chaos is amusing

Unpredictable games – strong nonlinearities:throw dice, play cards/chess, computer games ball games (football, soccer, basketball… impact)plus nonlinear rules (tennis 6/4,0/6,6/4, snooker)balancing (skiing, skating, kayak, surfing,…)

Ice-hockey (one of the most unpredictable games)- impacts between club/puck/wall- impacts between players/wall - self-balancing of players on ice (non-holonomic)- continuous and fast exchanging of players

Stabilization (balancing)

Control force:Q = – Px – Dx

Large delays can destroy this simple strategy, buttime-periodic parameters can help…

.

Balancing inverted pendulum

Higdon, Cannon (1962) …10-20 papers / year

n = 2 DoF , x ; x – cyclic coordinate

linearization at = 0

Qml

mgl

xmml

mlml 0

sin

sin

cos

cos22

21

21

21

212

31

cos6sin6)2sin(2

3)cos34( 22

ml

Q

l

g

Qmll

g 66

Human balancing

Analogous or digital?Winking, eye-motion – ‘self-sampling’plus neurons firing… still, not ‘digital’

1) Q(t) = P(t) + D(t) (PD control)

≡ 0 is exponentially stable D > 0, P > mg

2) Q(t) = P(t – ) + D(t – ) (with ‘reflex’ delay )

Qmll

g 66

0)(66

mgPml

Dml

.

.

0)(6

)(6

)(6

)( tl

gtP

mltD

mlt

0)()()()( 2

n ttptdt

0Re0 ,...2,12n

2 peed

Schurer Math Nachr 1948 … Stepan Ret Dyn Syst 1989… Sieber Krauskopf Phys D 2004

Stability chart & critical delay

instabilityg

lcr 3

]m[3.0l

0

2/

Stability chart & critical reflex delay

instabilityg

lcr 3

]s[1.0)103(3.0

f

20

]Hz[5.24

1

0

2/

]m[3.0l

Experimental observations

Kawazoe (1992)untrained manual control

(Dagger, sweep, pub)

Self-balancing:Betzke (1994)target shooting0.3 – 0.7 [Hz]

(Daffertshofer 2009)

Stability is the art of keeping the balance

2cr

T

Labyrinth – human balancing organ

Both angle and angular velocity signals are needed!

Dynamic receptor

Static receptor

Vision and balancing

• Vision can help balancing even when labyrinth does not function properly (e.g., ‘dry ear’ effect)

• The visual system also provides the necessary angle and angular velocity signals!

• But: the vertical direction is needed (buildings, trees), otherwise it fails…

• Delay in vision and ‘thinking’

Tactile / auditory / visual ~ sensors / cortexorgan

effectoverall

performance

cortexbrain

smalllarge

skinpressure

smallsmall

object

fast

mediumsmall

earsound

mediummedium

medium

largesmall

eyelight

smalllarge

slow

delaydistance

delaydistance

Lynx ~ Italian (National) Academy

Colliculus superior

eyes

brain

arm

MTLτ > 0.6 s

τ ~ 0.1sMedial Temporal Loop

Human balancing – some conclusionsWe could reduce the delay below critical value

through the MTL (Medial Temporal Loop)

But we cannot reduce much the thresholds of our sensory system (glasses...)

Both delay and threshold increase with age – see increasing number of fall-overs in elderly homes

Reduce gains, add stochastic perturbation to signal to decrease threshold at a 3rd sensory system – our feet (Moss, Milton, Nature, 2003)

Delay & threshold lead to chaos… (stochastic nature)

Digital balancing

1) Q = 0 – no control

= 0 is unstable

2) Q(t) = P(t) + D(t) (PD control)

= 0 is exponentially stable D > 0, P > mg

3) Q(t) ≡ P (tj – ) + D (tj – ) (with sampling )

Qmll

g 66

06 l

g

0)(66

mgPml

Dml

.

.

,...2,1),,[),[ 1 jttttt jjjj

Alice’s Adventures in Wonderland

Lewis Carroll (1899)

Sampling delay of digital control

delay ZOH

Digitally controlled pendulum

,

jutl

gt )(

6)( ),[ jj ttt

)()(6 jjj tPtD

mlu ,2,1, j ))(())((

6ttPttD

mlu j (Claussen)

Stability of digital control – sampling

Hopf

pitchfork

l

g6

jj Axx 1

j

j

jj

u

t

t

)(

)(

x

066

shchsh

1chshch

2

2

Dml

Pml

A

0)(det AI11,2,3

cr2

53ln

6g

l

ABB

Sampling frequency of industrial robots ~ 30 Hz for the years 1990 – 2005 above 100 Hz recently

Force control (EU 6FP RehaRob project),and balancing (stabilization-)tasks

RehaRob Balancing

Random oscillations of robotic balancing

sampling time and

quantization (round-off)

Stability of digital control – round-off

h – one digit converted to control force

det(I – B) = 0 1

= e >1, 2 = e–, 3

= 0

h

tPtDh

mlu jj

j

)()(int

6

lg

jjj

/6

)(1

xgBxx

000

chsh

chsh

1chsh2

B

jh

DjhP

ml

j

xxh 216 int

0

0

)(2

xg

1D cartoon – the micro-chaos map

Drop 2 dimensions, rescale x with h a e, b P

A pure math approach ( p > 0 , p < q )

solution with xj = y(j) leads to -chaos map,

a = ep, b = q(ep – 1)/p a > 1, (0 <) a – b < 1

small scale: xj+1= a xj , large scale: xj+1= (a – b) xj

)int(int)()( tyqtpyty

)int(1 jjj xbaxx

Micro-chaos map

large scale

small scale

Typical in digitallycontrolled machines

)int(1 kkk xbaxx

2D micro-chaos map

ZOH + delay, and round-off for 1st order process:

(p > 0, p < q)

Solution and Poincare lead to

(a >1, a – b < 1)

Linearization at fixed points leads to eigenvalues

So in 1 step the solution settles at an attractor that has a graph similar to the 1D micro-chaos map

1)int(int)()( tyqtpyty

)int( 11 jjj xbaxx

)1(,0)int(

0

0

1021

1

1

1

a

xbx

x

ax

x

jj

j

j

j

Csernak,Stepan (Int J Bif Chaos ’09)

3D micro-chaos

Enikov,Stepan (J Vib Cont, 98)

Vertical direction?

Segway – mechanical model

M

M

lm

glm

xmmlm

lmlm

Rr

r

rwr

rr

12221

21

23

21

212

31

sin

sin

cos

cos

2/l

wm

bm

MM

R

k

Lm

k

gmq 00 sin

qDPqM

3210 DPDPM

accelerometer

x

Segway control with delay

Analog case

Advance DDE …unstable for any “time delay”.

Digital case

0)()()(

)()()(

000103

032n

tptdtp

tdtt

0))(())(())((

))(()()(

013

32n

ttpttdttp

ttdtt

0

0rh

,)( 0 jhrhtttt j ),[ 1 jj ttt

Retarded DDE

Analog (Hayes, 1951) Digital)()()( tbxtaxtx )()()( rhtbxtaxtx j

,...1,0, jjht j

),[ 1 jj ttt

jj yy 1

0100

0010

0001

)1e(00e

,

)(

)(

)(

)(

2

1

1

ahabah

rj

j

j

j

j

tx

tx

tx

tx

y

0 bea 1rh

Neutral DDE

Analog (Kolmanovski, Nosov 1986) Digital)()()( txbtaxtx )()()( rhtxbtaxtx j

,...1,0, jjht j

),[ 1 jj ttt

jj zz 1

0100

0010

e00e

)1e(00e

,

)(

)(

)(

)(

11

ahah

ahabah

rj

j

j

j

j

ba

tx

tx

tx

tx

y

0 eba 1rh

Advanced DDE

Analog (El’sgolt’c 1964) Digital)()()( txbtaxtx )()()( rhtxbtaxtx j

,...1,0, jjht j

),[ 1 jj ttt

jj ww 1

0100

0010

e00e

)1e(00e

,

)(

)(

)(

)(2

11

ahah

ahabah

rj

j

j

j

j

aba

tx

tx

tx

tx

w

02 eba1rh

)()()( 1 txtxtx bba

02 eaeb0Re ,...2,1

Balancing the self-balanced

Warning: only fathers have the right to do this…

Thank you for your attention!

Delay effects in brain dynamics Phil. Trans. R. Soc. A 367 (2009) doi: 10.1098/rsta.2008.0279

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