global analysis of limit cycles in networks of oscillators ...stan/slidesnolcos2004_beamer.pdf ·...
Post on 12-Mar-2020
19 Views
Preview:
TRANSCRIPT
Goals and motivationThe dissipative oscillator
Global analysis of PLSPL Fitzhugh-Nagumo oscillator
Conclusions
Global Analysis of Limit Cycles in Networks ofOscillators
Guy-Bart STAN, Rodolphe SEPULCHRE
Guy-Bart Stan
University of Liege
Nolcos, September, 3, 2004
−0.8−0.6
−0.4−0.2
00.2
0.40.6
−1
−0.5
0
0.5
1−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
X
ξ
G.-B. Stan- ULg Nolcos 2004 [1/14]
Goals and motivationThe dissipative oscillator
Global analysis of PLSPL Fitzhugh-Nagumo oscillator
Conclusions
Outline of the talk
1 Goals and motivation2 The dissipative oscillator3 Global analysis of PLS4 PL Fitzhugh-Nagumo oscillator5 Conclusions
G.-B. Stan- ULg Nolcos 2004 [2/14]
Goals and motivationThe dissipative oscillator
Global analysis of PLSPL Fitzhugh-Nagumo oscillator
Conclusions
Goals and motivation
Main theme: Global analysis and control of sustainedoscillations (limit cycles) in networks of oscillators.Based upon:
high-dimensional generalizations of the Van der Pol andFitzHugh-Nagumo equations,passivity and related concepts,application of input-output stability theory to the analysis oflimit cycles.
Goal: Definition of a class of oscillators whose orbital stabilityproperties extend directly to networks of oscillators.Motivation:
Bridge the gap between equilibrium points and limit cyclesanalysis tools in nonlinear systems,Control of amplitude, frequency and phase of oscillations innetworks of oscillators.
Applications:
robotics: rhythmic tasks robots (i.e. walking robots, jugglingrobots, dexterous robots),biology: fundamental oscillation mechanism.
G.-B. Stan- ULg Nolcos 2004 [3/14]
Goals and motivationThe dissipative oscillator
Global analysis of PLSPL Fitzhugh-Nagumo oscillator
Conclusions
CharacterizationAnalytical results for k & k∗
The dissipative oscillator - Characterization
u yO
Internal description: ’global’ limit cycleExternal description: a dissipation inequality, suitable foranalysis of interconnectionsIncludes prototype oscillators as simplest examples (Van derPol, Fitzhugh-Nagumo) but not restricted to low-dimensionalmodels
G.-B. Stan- ULg Nolcos 2004 [4/14]
Goals and motivationThe dissipative oscillator
Global analysis of PLSPL Fitzhugh-Nagumo oscillator
Conclusions
CharacterizationAnalytical results for k & k∗
Analytical results for k & k∗
−k
−
passiveu y
Σ
φk(y)
A parameter (k) controls the negative slope of the staticnonlinearity φk(y) = −ky + φ(y), with φ(·) a stiffeningnonlinearity, satisfying yφ(y) > 0, ∀y 6= 0 (e.g. cubic)Absolute stability for k = 0; equilibrium loses stability atk = k∗ ≥ 0Bifurcation codimension (one or two) dictates one of twoscenarios for global oscillations for k & k∗
G.-B. Stan- ULg Nolcos 2004 [5/14]
Goals and motivationThe dissipative oscillator
Global analysis of PLSPL Fitzhugh-Nagumo oscillator
Conclusions
CharacterizationAnalytical results for k & k∗
Analytical results for k & k∗
−k
−
passiveu y
Σ
φk(y)
Bif. codim. = 2 → Hopf bifurcation : GAS\Es(0) limit cyclefor k & k∗.Bif. codim. = 1 → pitchfork bifurcation, global bistability
G.-B. Stan- ULg Nolcos 2004 [5/14]
Goals and motivationThe dissipative oscillator
Global analysis of PLSPL Fitzhugh-Nagumo oscillator
Conclusions
CharacterizationAnalytical results for k & k∗
Analytical results for k & k∗
y
−−
Σ
R
adaptation
1τs+1
φk(y)
u
global bistability + feedback adaptation loop : GAS\Es(0)limit cycle for k & k∗.
G.-B. Stan- ULg Nolcos 2004 [5/14]
Goals and motivationThe dissipative oscillator
Global analysis of PLSPL Fitzhugh-Nagumo oscillator
Conclusions
CharacterizationAnalytical results for k & k∗
Analytical results for k & k∗
−k
−
passiveu y
Σ
φk(y)
y
−−
Σ
R
adaptation
1τs+1
φk(y)
u
What about a particular value of k > k∗?
G.-B. Stan- ULg Nolcos 2004 [5/14]
Goals and motivationThe dissipative oscillator
Global analysis of PLSPL Fitzhugh-Nagumo oscillator
Conclusions
Definition of a PL dissipative oscillatorNumerical results for a particular value of k > k∗, n = 2Upper bounds on the switching times, n = 2
Global analysis of PLS
RESULT : global stability analysis of limit cycles of PLS canbe proven using quadratic stability of Poincare maps(Goncalves (2000))HOW : express Poincare maps induced by an LTI flowbetween 2 switching surfaces as linear transformationsanalytically parameterized by a scalar function of the state.IDEA : compute quadratic Lyapunov functions forPoincare maps.The search for Lyapunov functions is done by solving a setof LMIs.
G.-B. Stan- ULg Nolcos 2004 [6/14]
Goals and motivationThe dissipative oscillator
Global analysis of PLSPL Fitzhugh-Nagumo oscillator
Conclusions
Definition of a PL dissipative oscillatorNumerical results for a particular value of k > k∗, n = 2Upper bounds on the switching times, n = 2
Poincare maps for PLS
T1
limit cycle
T2
S0
3
eq. point
eq. point
v
v
v
vx∗
1
x∗
2
vx∗
0
S1
G.-B. Stan- ULg Nolcos 2004 [7/14]
Goals and motivationThe dissipative oscillator
Global analysis of PLSPL Fitzhugh-Nagumo oscillator
Conclusions
Definition of a PL dissipative oscillatorNumerical results for a particular value of k > k∗, n = 2Upper bounds on the switching times, n = 2
Individual Poincare maps (Impact maps)
S0
V0(·)
x∗0
∆0
x∗0 + ∆0
x = Ax − By = Cx
T1S1
V1(·)
x∗1 + ∆1
∆1
x∗1
S1 = {x ∈ Rn : Cx + d = 0}
T1 : S0 → S1
x∗
1 = T1(x∗
0 )
or
∆1 = T1(x∗
0 + ∆0) − x∗
1
= H(t)∆0
Quadratic stability of impact maps
r(t) = ∆T0 P(t)∆0
= V0(∆0) − V1(H(t)∆0) > 0, ∀t ∈ [tmin, tmax ]LMIs−−−→ [ t1
︸︷︷︸
=tmin
, t2, . . . , tN︸︷︷︸
=tmax
]
G.-B. Stan- ULg Nolcos 2004 [8/14]
Goals and motivationThe dissipative oscillator
Global analysis of PLSPL Fitzhugh-Nagumo oscillator
Conclusions
Definition of a PL dissipative oscillatorNumerical results for a particular value of k > k∗, n = 2Upper bounds on the switching times, n = 2
Definition of a PL dissipative oscillator
Σ is a linear systemFeedback static nonlinearity : φk(y) = −ky + y3 → fpls(y)
−1.5 −1 −0.5 0 0.5 1 1.5−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
y
y3 −k py
Cubic nonlinearity (blue); piecewise linear approximation (red)
−m
m
y
fpls(y)
py − d
py + d
−m
m
km
m =√
k3
d = m(p + k)
NLS: Analytical results for k & k∗ PLS: Numerical results for k > k∗
G.-B. Stan- ULg Nolcos 2004 [9/14]
Goals and motivationThe dissipative oscillator
Global analysis of PLSPL Fitzhugh-Nagumo oscillator
Conclusions
Definition of a PL dissipative oscillatorNumerical results for a particular value of k > k∗, n = 2Upper bounds on the switching times, n = 2
PL dissipative oscillator numerical analysis
S0 S1
m−m
x∗0
−x∗0
x∗1
y = Cx
−x∗1
(R3)(R1)
∆0
x = A1x
(R2)
x = A2x − dB x = A2x + dB
V0(·)V1(·)
∆1 = H(t)∆0
A1 is not Hurwitz, A2 is Hurwitz, m > −CA−12 dB
1 Existence of limit cycles
f1(t∗
1 , t∗2 ) = Cx∗
0 (t∗1 , t∗2 ) + m = 0
f2(t∗
1 , t∗2 ) = Cx∗
1 (t∗1 , t∗2 ) − m = 0
2 Quadratic stability of impact maps
r(t) = ∆T0 P(t)∆0
= V0(∆0) − V1(H(t)∆0) > 0, ∀t ∈ [tmin, tmax ]LMIs−−−→ [ t1
︸︷︷︸
=tmin
, t2, . . . , tN︸︷︷︸
=tmax
]
G.-B. Stan- ULg Nolcos 2004 [10/14]
Goals and motivationThe dissipative oscillator
Global analysis of PLSPL Fitzhugh-Nagumo oscillator
Conclusions
Definition of a PL dissipative oscillatorNumerical results for a particular value of k > k∗, n = 2Upper bounds on the switching times, n = 2
Upper bounds on the switching times, n = 2
Generic assumption: A1 has no real unstable eigenvalue.
1 (R2), A1 is not Hurwitz
Generically, A1 has stable and unstable eigenspaces: switching time is unboundedn = 2 : A1 is anti-Hurwitz (⇔ no stable eigenspace): tmax is given by the worst switching scenario(Cx0 = −m, CAx0 = 0)
2 (R1) and (R3): A2 is Hurwitz
guarantee contraction in (0, timax)
prove that contraction holds ∀t > timax
G.-B. Stan- ULg Nolcos 2004 [11/14]
Goals and motivationThe dissipative oscillator
Global analysis of PLSPL Fitzhugh-Nagumo oscillator
Conclusions
Simulation results
PL Fitzhugh-Nagumo oscillator
−−
G
1τs+1
y
fpls(·)
u
0 2 4 6 8 10 12 14 16 180.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Oscillator with ki=1, k
p=1.200000e+000 and p=5
Min
(eig
(Ri(t
)))
with
impl
emen
tatio
n of
the
coni
c re
latio
n
Time (s)
R1
R2
G (s) =1
s + 1, τ = 20, k∗ = 1 +
1
τ= 1.05 < 1.2 = k
t∗1 = 8.88, t∗2 = 8.4
tmax(R2)= 9
G.-B. Stan- ULg Nolcos 2004 [12/14]
Goals and motivationThe dissipative oscillator
Global analysis of PLSPL Fitzhugh-Nagumo oscillator
Conclusions
Simulation results
PL Fitzhugh-Nagumo oscillator
−−
G
1τs+1
y
fpls(·)
u
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
y
ξ
State space for one oscillator with ki=1, k
p=1.200000e+000 and p=5
G (s) =1
s + 1, τ = 20, k∗ = 1 +
1
τ= 1.05 < 1.2 = k
t∗1 = 8.88, t∗2 = 8.4
tmax(R2)= 9
G.-B. Stan- ULg Nolcos 2004 [12/14]
Goals and motivationThe dissipative oscillator
Global analysis of PLSPL Fitzhugh-Nagumo oscillator
Conclusions
Conclusions
Global analytical results for k & k∗ and n > 2 : dissipativeoscillator;
Global numerical results for a particular value of k > k∗ andfor n = 2 : PL dissipative oscillator;
Other results (not presented): interconnection,synchronization of dissipative oscillators.
G.-B. Stan- ULg Nolcos 2004 [13/14]
Goals and motivationThe dissipative oscillator
Global analysis of PLSPL Fitzhugh-Nagumo oscillator
Conclusions
That’s all !
Questions...
G.-B. Stan- ULg Nolcos 2004 [14/14]
top related