Partial Delayed Feedback Control andits DSP Implementation

T. Ueta†, Y. Toyosaki†, S. Tsuji†, and T. Kousaka‡
Tokushima University, Fukuyama University

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Background
Controlling chaos — Stabilizing an unstable
periodic orbit in given nonlinear dynamical system
OGY method• pole assignment technique• delayed feedback control (DFC)• external force control (EFC)
•









K. Pyragas, “Continuous control of chaos by selfcontrolling feedback,” Phys. Lett, 170A, pp. 421–428, 1992

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DFC — Delayed feedback control
Control input: a difference between
the present state and the τ-delayed stateKu(t)+ x(t−τ)−DChaotic Systemx(t)MWSCAS2004@Hiroshima – p.3/39

Background
Delayed Feedback Control(DFC)
Merit:• Simple structure• No information about UPOs needed• Can be a UPO detector
Demerit:• τ — unknown for autonomous systems• infinite dimension• difficult to analyze its stability
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Hybrid systems
Dynamical systems including interrupts,switches, impulsive inputs• showing chaotic behavior frequently•

Controlling chaos in hybrid systems:






T. Kousaka, S. Tahara, T. Ueta, M. Abe, and H.
Kawakami, “Chaos in a simple hybrid system andits control,” IEE Electronics Letters, Vol.37, No.1,
pp.1 − 2, Jan.2001

using an OGY-like method.
DFC for chaotic hybrid system is not given !

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In this talk...
Partial Delayed Feedback Control (PDFC)• PDFC by a digital signal processor (DSP)• EFC after DFC — stability analysis• possibly it is reasonable scheme for hybridsystems•

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Descriptions
A chaotic hybrid system:dx= f (t, x)
dt
where, x ∈ Rn , f : Rn → Rn
Assume that a solution x(t) = ϕ(t, x0 ) would bechaos.
A UPO:x0 = ϕ(0, x0 ) = ϕ(τ, x0 )where, τ is the period of the UPO.
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DFC
Conventional DFC
dx= f (t, x) + u(t)dt∀t > 0u(t) = K(x(t − T ) − x(t))

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PDFC
Partial Delayed Feedback Control(PDFC):dx= f (t, x) + u(t)dtifdx= f (t, x)dt

otherwise

C(x) ∈ M

u(t) = K(x(t − T ) − x(t))C(x): condition depending on the state.
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PDFC

S(t)

SWK+ x(t−τ)−Du(t)
Chaotic Systemx(t)MWSCAS2004@Hiroshima – p.10/39

PDFC
A control input computed from data stored in amemory is applied to the system intermittently.
• The control scheme may be “cutting corners,”“natural,” or “手抜き ” — “follow the system asis.”• possibly effective to: circuits having a restrictedtiming for control or a refractory period, robots
including passive joints, etc.
•MWSCAS2004@Hiroshima – p.11/39

Related works
P. Hovel and J. E. S. Socolar, “Stability domains
for time-delay feedback control with latency,”
Phys. Rev. E 68, 036206 (2003) The DFC input
latency is discussed.
• K. Myneni, T. A. Barr, N. J. Corron, and S. D.
Pethel, “New method for the control of fast
chaotic oscillations,” Phys. Rev. Lett. 83,2175-2178 (1999) Partial signal input is applied
as an open-loop control.
•MWSCAS2004@Hiroshima – p.12/39

PDFC for a simple chaotic system
T−vr+SWvAC

R S RS-FF
Q Q

BRc

R

E

T. Kousaka, et al., IEEJ Vol.122-C, No. 9, 2002.
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Behavior 1/4
T−vr+SWvAC

R S RS-FF
Q Q

B

Rc
E

R

A wave form of thecapacitor voltage v

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Behavior 2/4
switch position: AT−vr+SWv
AC

R S RS-FF
Q Q

B

RcRE

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Behavior 3/4
switch position: B
T−vr+SWvAC

R S RS-FF
Q Q

B

Rc

R

E

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Behavior 4/4
switch position: AT−vr+SWv
AC

R S RS-FF
Q Q

B

Rc

R

E

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Chaos: real parameters
T−

vr+SWvAC

R S RS-FF
Q Q

B

Rc

R

E

vr = 2.5[V], R = 51[kΩ], C = 1.0[µF], E = 3.0[V],pulse period: T = 30[msec].

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Circuit equation and simulation
dv= −(v − E) ifposition Adtdv= −v
otherwise position B
dt
32.5v(t)

21.5
10.5
000.5

11.5

2

horizontal: t[sec], vertical: v[V]MWSCAS2004@Hiroshima – p.19/39

Return map
2.5

vk+1→

2.0
1.5
1.0
1.0

1.5

2.0
vk →

2.5

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Apply PDFC to the circuit
T−vr+SWvACR S RS-FFQ Q

BRcRDSPMWSCAS2004@Hiroshima – p.21/39

Apply PDFC to the circuit
The control input is added into the terminal c.
• DSP stores sampled voltage sequence in the ringbuffer.
• Delay duration — τ = nT , n = 1, 2, . . . : pulse
period• Feedback gain K can be controlled by both DSP
and analog amplifier.
•MWSCAS2004@Hiroshima – p.22/39

Controlled systemdv
ifposition A= −(v − E − u(t))
dt
dv
= −v
otherwise position B
dtu(t) = S (v(t)) · K(v(t − T ) − v(t))
S (v(t)) =(1 ifposition A
0 otherwise position B

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Wave form of S (v(t))
S(v(t))

S10BABB A A BABABvv(t)002T4T6T

tu(t) = S (v(t)) · K(v(t − T ) − v(t)): not full energy ofK(v(t − T ) − v(t)) is consumed.
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K = 0.6, v(0) = 0.0, τ = 2T .
The control is enabled at t = 1.5[sec].
3
2.5v(t)

2
1.5
1
1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

0.4

u(t)

0.2
0−0.2
−0.4
1.2

tMWSCAS2004@Hiroshima – p.25/39

Basin of attractionv(0)10.00.0-10.0-1.0 0.0

0.25 0.85 3.25
K

5.75 6.3 6.5

The control is enabled from t > 1.53 [sec]White: u(t) converges, Black: u(t) otherwise.
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Basin of attraction
Detection of convergence for u(t):



0.025.5[sec]X













0.0

u (t )

t|u(t)| < ,

t=20.4[sec] ≈ 0.1.

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DSP implementation of PDFC

Texas Instruments, TMS320 C6711DSP +TMDX326040A
16-bit, 44 kHz, stereo I/O.
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DSP implementation of PDFC
Algorithm on DSP:
1. Allocation of a ring buffer whose size iscoincident to the sampling rate × τ.
2. AD conversion v(t) → x(t) and storing as 32bit
unsigned integer variable.
3. computing u = x(t − τ) − x(t) and output u.
Software: C language, CodeComposer v.2.0

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DSP implementation of PDFCMWSCAS2004@Hiroshima – p.30/39

DSP implementation of PDFCMWSCAS2004@Hiroshima – p.31/39

DSP implementation of PDFC
Physical measurement: v(t) and u(t), K = 0.3

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DSP implementation of PDFC
Start controlling

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DSP implementation of PDFC
Transition

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DSP implementation of PDFC
Transition

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DSP implementation of PDFC
Control completed

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Discussion
Robustness is obtained.
(confirmed by computing a basin of attraction)• DSP works well for PDFC in a hybrid system.
• PDFC (even though DFC): The system becomesan infinite dimensional system — too difficult toanalyze its stability•

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Featuring EFC
External Force Control:+K−x(t)UPO
datax(t)Chaotic System

Strategy:
1. Attract the orbit near an UPO by PDFC
2. Switch to EFC (very easy to implement it byprogram codes)
3. Stabilize UPO by EFC
4. Stability analysis is possible!

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Conclusions
PDFC—Partial Delayed Feedback Control• concept and examples
S(t)SW+ x(t−τ)
t• computer simulations and a DSP
K−implementation
D• PDFC + EFC u(t)Chaotic System
Future problemsx(t)• application to high-dimensional systems• stabilizing various UPOs
• stability analysis
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