Automatic threshold adjustment
for limit cycles holding a speciﬁed
stability in hybrid systems

Tetsushi Ueta and Yuu Miino
March 9, 2016
Tokushima University, Japanat NCSP2016, Waikiki, Honolulu, Hawaii

Hybrid systems

Hybrid systems—Finite automata
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eventx3

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mode3
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Phase event
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1Our original contributions
Computation of bifurcation sets in hybrid systems• T. Kousaka, T. Ueta, and H. Kawakami, “ Bifurcationof switched nonlinear dynamical systems,” IEEE
Trans. CAS-II, 46, no. 7, pp. 878–885, 1999.
• ...
• Y. Miino, D. Ito, and T. Ueta, “A computation method
for non-autonomous systems with discontinuous
characteristics,” Chaos, Solitons & Fractals, 77, 8, pp.
277–285, 2015.
2

Problem description

Given the hybrid system:dx= f(x)
dt
where, f contains several different dynamical systems
aligned by a threshold xth ∈ R
Assume that a solution is written by:
x(t) = φ(t, x0 ),x(0) = φ(0, x0 ) = x0 .

3Our specialityf0 (x) if x ∈ R1
f1 (x) if x ∈ R2dx=..
dt .


 f (x) if x ∈ Rm−1

⇔ φ0
⇔ φ1m−1⇔ φm−1

where Ri is a region of i.
∂φWe can evaluate ∂xthjj even though xthj areimplicit parameters!
D. Ito, T. Ueta, and K. Aihara: IEICE Trans. Fundum. JA-94,
No. 8, 2011. (Japanese)4Perturbations

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Poincaré mapping and its ﬁxed point

The ﬁxed point of the Poincaré map: T(x):T(x0 ) − x0 = 0(1)

The characteristic equation:
χ(µ) = det(DT(x0 ) − µI) = 06

Controlling chaos for hybrid systems

Controlling chaos by a threshold

D. Ito, et al. Int. J. Bifur. Chaos, 24, 10, 2014.
Threshold controller

7

Controlling chaos by a threshold
D. Ito, et al. Int. J. Bifur. Chaos, 24, 10, 2014.
• Controller perturbes the threshold value• The threshold value converges when completed8

In this talk, we propose...

Stability design

Solve a threshold value of the limit cycle whose
multiplier satisﬁes the speciﬁc stability.
Threshold value

DesiredmultiplierFeedforward andnon-dynamiccontroller

Oscillator(hybrid system)

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Open-loop control; an example
Computer-generated
threshold value−+Oscillatory
hybridsystem
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Almostuncontrollable

Thre

dshol

11

dsholThre

No affections
to the originalsystem

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Example: van der Pol oscillator with a threshold
Switched van der Pol oscillator:{dx
f0 (x, ϵ, ω)=dt
f1 (x, ϵ, ω)

if q(x) ⩽ 0
otherwise,(2)

where,(()

f0 =

yϵ0 (1 − x2 )y − ωx),f1 =yϵ1 (1 − x2 )y − ωx.

xth ∈ R is a threshold value, q(x) = x − xth ∈ R.
A solution:xi (t) = φi (t, xi0 ),xi (0) = xi0 .

(3)13

Limit cycles of hybrid dynamical systems

Poincaré section

The threshold value may change the stability of thecycle.

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Sample phase portrait

ϵ0 = 0.2, ϵ1 = 0.1, and xth = 0.1.
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Bifurcation diagram
1
0.9
0.8
0.7
0.6
0.5
0.5

0.6

0.7

0.8

0.9

116

Derivative of TThe derivative with the initial value of the Poincaré map:
∏ ∂T1−i∂T=∂x0∂x1−imi=0

Each Jacobian matrix is given by:.

(4)

t=τ1−i

∂Ti 1∂q  ∂φi= In −fi ∂q∂xi∂x ∂xi
· fi
∂x(5)

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hoge

Π0
Π1{}
dx
=
(x, y) | q(x) = 0,>0 ,dt{}dx=(x, y) | q(x) = 0,<0 .
dt(6)The local map is written by
T 0 : Π0 → Π 1 ;T1 : Π 1 → Π 0 ;x0 7→ x1 = φ0 (τ0 , x0 ),x1 7→ φ1 (τ1 , x1 ).

(7)

The Poincaré map:
T(x0 ) = T1 ◦ T0 (x0 ).

(8)18

Local coodinate
The local coordinate: u ∈ Σ0 ⊂ R
A projection p and an embedding map p−1 :p−1 : Σ0 → Π0 , p : Π0 → Σ0 .

(9)

The Poincaré mapping on the local coordinate:
Tℓ : Σ0 → Σ0 ;u 7→ p ◦ T ◦ p−1 (u).

(10)

The ﬁxed point of the Poincaré mapping u∗
Tℓ (u0 ) − u0 = 0.

(11)
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Jacobian matrix
The Jacobian matrix is given by∂Tℓ∂p ∂T ∂p−1
= DTℓ (u0 ) =.
∂u0∂x ∂x0 ∂u(12)

The characteristic equation for the ﬁxed point is givenbyχℓ (µ) = det (DTℓ − µ∗ I) = 0,(13)where µ∗ is speciﬁed multiplier. We can obtain aparameter xth with the speciﬁed µ∗ by solving Eq. (11)and (13).
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Computation of the ﬁxed point

The derivative with xth is required in Newton’s method
and is obtained from the previous study as follows:∂Tℓ∂p ∂T=.
∂xth∂x ∂xth(14)Now we are ready to solve them for u0 and xth .

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Control procedure

1. Give the desired multiplier
2. Compute the threshold value satisfying themultiplier
3. Change the threshold value

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Isocline, ϵ0 = 0.2
2(a)
(b)
(c)
(d)
(e)
(f )

1.5

10.50

ϵ1 = 1.0
ϵ1 = 0.4
ϵ1 = 0.3
ϵ1 = 0.2
ϵ1 = 0.1
ϵ1 = 0.0

Er ⇔ xth−0.5−1−1.5−2

00.2

0.4

0.6

0.8

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Isocline, ϵ0 = 0.2
2

1.5

Thresholdsetting =control

1

0.5

0

−0.5

−1

−1.5

−2

00.2

0.4

0.6

0.8

124

At point A, µ = 0.4
4
32
10−1−2−3−4−4−3−2−10123425

Features and advantages• Change of the threshold value: an implicit
parameter• No control energy is consumed• No explicit parameter is changed — system
unchanged• Transition: almost original system behavior• Non-dynamical controller is realized

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Concluding Remarks• Stability with the threshold value for the given
hybrid systems are computable• A limit cycle holding a desirable stability iscontrolled by the threshold value• The control scheme is easy: just change a thresholdvalue stepwisely• NO TRANSITION, No affection to the state variables,
basically.

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ReferencesK. Aihara, J. Imura and T. Ueta (eds), Analysis andControl of Complex Dynamical Systems, Springer,Tokyo, Mar. 2015.
D. Ito, T. Ueta, T. Kousaka, J. Imura, and K. Aihara,“Controlling Chaos of Hybrid Systems by Variable
Threshold Values,” Int. J. Bifurcation and Chaos, Vol.
24, No. 10, 1450125, 2014.
T. Ueta, S. Tsuji, T. Yoshinaga, and H. Kawakami,“Calculation of the isocline for the ﬁxed point with aspeciﬁed argument of complex multipliers,” in Proc.
IEEE/ISCAS 2001, Vol. 2, pp. 755–758, 2001.
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AcknowledgementThis work has been supported by KAKENHI 25420373.

The authors also would like to thank Prof. Kawakami(Prof. Emeritus, Tokushima University) for his valuableadvice regarding this work.

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