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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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