PASSIFICATION BASED CONTROLLED SYNCHRONIZATION OF … · 2012-09-06 · PASSIFICATION BASED...

PASSIFICATION BASED CONTROLLED SYNCHRONIZATION

OF COMPLEX NETWORKS

Alexander Fradkov

Institute of Problems in Mechanical Engineering Saint Petersburg, Russia

______________________________________________________ INRIA, Lille, Sept.5, 2012

Institute of Problems in Mechanical Engineering of RAS Control of Complex Systems Laboratory

Outline

1. Introduction. 2. Control of dynamical networks: problems 3. Control of dynamical networks: models 4. Controlled synchronization of dynamical

networks: results 5. Conclusions

Institute of Problems in Mechanical Engineering of RAS Control of Complex Systems Laboratory

It is convenient to split history of automatic control into three most relevant periods: ‘deterministic’,

‘stochastic’ and ‘adaptive’. Ya.Z.Tsypkin. Adaptation and learning in automatic systems. Moscow: Nauka, 1968 (New York: Academic Press, 1971)

Institute of Problems in Mechanical Engineering of RAS Control of Complex Systems Laboratory

ХXI century: ‘networked’ period Search over Web of Science: number of papers with ‘control’ AND

‘network’ is doubled every 5-6 years ва статей в рецензируемых журналах по данной тематике за 5-6 лет:

Institute of Problems in Mechanical Engineering of RAS Control of Complex Systems Laboratory

Examples of controlled networks

Distributed factories Networks of mobile robots (production, transportation, finance networks)

Institute of Problems in Mechanical Engineering of RAS Control of Complex Systems Laboratory

Electric power networks Pogromsky, A.Yu., A.L. Fradkov and D.J. Hill, Passivity based damping of power system oscillations. 35th IEEE Conf. Decision and Control, Kobe, 1996, pp.3876-3881. D. J. Hill and G. Chen, “Power systems as dynamic networks,” IEEE Int. Symp. On Circuits and Systems, Kos, Greece, 2006, 722–725.

sin( )i

i i i i i j ij i j ij C

M D VV b Pθ θ θ θ∈

+ + − =∑

Institute of Problems in Mechanical Engineering of RAS Control of Complex Systems Laboratory

Control of complex crystalline lattices Aero E., Fradkov A.L., Andrievsky B., Vakulenko S. Dynamics

and control of oscillations in a complex crystalline lattice. Physics Letters A., 2006, V. 353 (1), pp 24-29.

Ecological networks Pchelkina I., Fradkov A.L. Control of oscillatory behavior of multi-species

populations. Ecological Modelling 227 (2012) 1– 6.

Energy/ entropy production function:

Multi-species Lotka-Volterra model:

Control goal: W(X(t)) W*

+=

+⋅+⋅=

=

⋅+⋅=

∑

∑

=

−

=

−

.,..1,)()()(

,,..2,1,)()(

1

1

1

1

NMltutxaktxdtdx

Mitxaktxdtdx

l

N

jjljlll

l

N

jjijiii

i

β

β

.log)(1∑=

−=

N

i i

i

i

iii n

xnx

nxW β

Institute of Problems in Mechanical Engineering of RAS Control of Complex Systems Laboratory

9

COOPERATIVE CONTROL OF ROBOTS

Football of robots

Institute of Problems in Mechanical Engineering of RAS Control of Complex Systems Laboratory

What is control of network?

ХХ century: MIMO systems ХХI century: networks

Institute of Problems in Mechanical Engineering of RAS Control of Complex Systems Laboratory

Plant

Controller

C C

C C

C

C

Control of network: problem statement Consider interacting subsystems (agents) - state vectors of agents, - inputs (controls), - measurable outputs Control goal: synchronization (consensus) for outputs: P-controller (consensus protocol ): is set of neighbors for th agent

N

1( , ) ( , )

n

i i i ij ij i jj

x F x u a x xϕ=

= +∑ ( )i i iy h x=

( )ix t ( )iu t( )iy t

lim ( ) ( ) 0, , 1,...,i jxy t y t i j N

→∞− = =

( ), 1,...,i

i j ij N

u K y y i N∈

= − =∑

iN i

Institute of Problems in Mechanical Engineering of RAS Control of Complex Systems Laboratory

Control of network: special case

– Laplace matrix.

Information graph G:

1,

1

11 12 1

21 22 21,

1 2

( ) a ( ) , 1,...,

( ) a , 1,...,

a a aa aa a a

a a a

N

i i ij j i ij j i

N

i i ij j ij

N N

ii ijNj j i

N N NN

x f x c x x u i N

x f x c x u i N

A

= ≠

=

= ≠

= + Γ − + =

= + Γ + =

= − =

∑

∑

∑

a 0ij >

L A= −

:arc j i∃ →

Institute of Problems in Mechanical Engineering of RAS Control of Complex Systems Laboratory

Networked controller

• - matrix of interactions in controller, • - Laplace matrix of controller. • Control goal – consensus:

Network control of network: - closed loop Laplace matrix

B1,

( ), 1,...,N

i ij j ij j i

u b x x i N= ≠

= Λ − =∑

L B= −

lim ( ) ( ) 0, , 1,...,i jxx t x t i j N

→∞− = =

( )L A B= − +

Institute of Problems in Mechanical Engineering of RAS Control of Complex Systems Laboratory

Laplace matrix properties: • Defect of L = number of graph connected components (λ1=0) • L is symmetric graph is undirected (λ2 is algebraic connectivity) • Spectrum of L is in the closed right hand part half-plane • For digraph defect L =1 directed spanning tree exists Agaev-Chebotarev theorem (Aut.Rem.Control, 9, 2000, LAA, 2005): Defect of L = min. number of trees in spanning forest

Institute of Problems in Mechanical Engineering of RAS Control of Complex Systems Laboratory

Existing results Consensus conditions - I

, xi R1 Consensus exists directed spanning tree exists

The states of the agents tend to the average in initial conditions

(W. Ren and R.W. Beard, “Consensus seeking in multiagent systems under dynamically changing interaction topologies,” IEEE Trans. Automat. Contr., vol. 50, no. 5, pp. 655–661, 2005.)

X=(x1, … ,xN )T , dX/dt = - KLX

, , 1,...,i i i ix u y x i N= = = ∈

Institute of Problems in Mechanical Engineering of RAS Control of Complex Systems Laboratory

Consensus conditions - II Consensus in the networks of 2nd order agents (W. Yu, G.Chen, M.Cao. On second-order consensus in multi-agent dynamical

systems with directed topologies and time delays IEEE Conf. Dec.Contr. (CDC-2009), Shanghai Dec.2009.)

Consensus Directed spanning tree exists and the inequality holds:

- eigenvalues of Laplace matrix =ℜ( ) +jℑ( )

1 1

( ) ( )

( ) ( ) ( ), 1,2,...,

i iN N

i ij j ij jj j

x t v t

v t L x t L v t i Nα β= =

=

= − − =∑ ∑

2 2

2 22

( )max( ( ) ( ) )

i

i Ni i i

β µα µ µ µ≤ ≤

ℑ>

ℜ ℜ +ℑ iµ iµ iµ iµ

Institute of Problems in Mechanical Engineering of RAS Control of Complex Systems Laboratory

Consensus conditions - III Consensus in the networks of 2nd order agents with delay (W. Yu, G.Chen, M.Cao. IEEE CDC-2009, Shanghai Dec.2009)

Let directed spanning tree exists and the following inequalities hold:

[ ]

1 1

22

2 22

10 12

1

2 4 22 4 21

1 121

( ) ( )

( ) ( ) ( )

( )max( ( ) ( ) )

min , 0 2

4( ) ( )cos

2

i iN N

i ij j ij jj j

i

i Ni i i

iii N

i

i i ii i ii i

i

x t v t

v t L x t L v t

where

α τ β τ

µβα µ µ µ

θτ τ θ πω

µ β + µ β µ αµ α µ ω βθ ω

ω

= =

≤ ≤

≤ ≤

=

= − − − −

ℑ>

ℜ ℜ +ℑ

< = < <

+ℜ −ℑ= =

∑ ∑

Institute of Problems in Mechanical Engineering of RAS Control of Complex Systems Laboratory

18

Controlled synchronization of networks

1,( ) a ( ) , 1,...,

N

i i ij j i ij j i

x f x c x x u i N= ≠

= + Γ − + =∑

Most existing results deal with fully controlled agents, especially in adaptive control…

Challenge: To design decentralized

adaptive output feedback control ensuring synchronization

under conditions of uncertainty and incomplete control

19

Institute for Problems of Mechanical Engineering of RAS Control of Complex Systems Laboratory

Controlled network of linear agents:

dxi/dt=Axi+Bui , yi =CTxi

Output feedback (Diffusion coupling, Consensus protocol):

( ), 1,...,i

i j ij N

u K y y i N∈

= − =∑

W(s)=CT(sI-A)-1B – transfer matrix

lim ( ) ( ) 0, , 1,...,i jxx t x t i j N

→∞− = =Control goal:

20

Institute for Problems of Mechanical Engineering of RAS Control of Complex Systems Laboratory

Passivity (Willems, 1972) dx/dt=f(x)+g(x)u , y=h(x)

There is a function V(x) ≥ 0 (storage function):

For linear systems storage function can be quadratic: V(x)=xTPx, P=PT>0 ( PA+A TP<0, PB=CT )

0

( ( )) ( (0)) ( ) ( )t

TV x t V x y u dτ τ τ≤ + ∫

21

Passification theorem (Fradkov: Aut.Rem.Contr.,1974(12), Siberian Math.J. 1976, Europ. J.Contr. 2003; Andrievsky, Fradkov, Aut.Rem.Contr., 2006 (11) ) Let W(s)=CT(sI-A)-1B – transfer matrix of a linear

system. The following statements are equivalent: A) There exist matrix P=PТ >0 and row vector K, such that PAK+AK

T P<0, PB=(CGT), AK=A+BKCT B) Matrix GW(s) is hyper-minimum-phase (det[GW(s)] has Hurwitz numerator, GCTB=(GCTB)T >0 ) C) There exists K such that feedback u=Ky+v renders the system strictly passive «from input v to output Gy »

Institute for Problems of Mechanical Engineering of RAS Laboratory “Control of Complex Systems”

22

Institute for Problems of Mechanical Engineering of RAS Control of Complex Systems Laboratory

Theorem (Junussov, Fradkov, Aut. Rem.Contr., 2011, 8). A1. Network graph is balanced and has directed spanning tree. A2. gW(s) is hyper-minimum-phase for some g=(g1,…gn). (det[gW(s)] has Hurwitz numerator, gCTB=(gCTB)T >0 ) Let K=µg, where Then synchronization is achieved and

2/µ κ λ> inf ( ( ))R

e gW iω

κ ω∈

= ℜ

1lim( ( ) (1 ) (0)) 0, 1,..., .At Ti d nt

x t d e I x i d−

→∞− ⊗ = =

23

Institute for Problems of Mechanical Engineering of RAS Control of Complex Systems Laboratory

Example. Network of double integrators, N=4.

24

Institute for Problems of Mechanical Engineering of RAS Control of Complex Systems Laboratory

Example. Network of double integrators, N=4.

Simulation results.

μ=1

μ=0.7

μ=0.5

μ=0.3

25

Institute for Problems of Mechanical Engineering of RAS Control of Complex Systems Laboratory

Extension 1 (Adaptive control): 1. Network graph with a spanning tree 2. Adaptive controller:

- tunable parameters

Adaptation algorithm:

where

26

Institute for Problems of Mechanical Engineering of RAS Control of Complex Systems Laboratory

Extension 2 (Fradkov, Junussov, IEEE Conf. Dec.Contr., Orlando, Dec. 2011): 1. Network digraph with a directed spanning tree 2. Nonlinear (Lurie) models of agent dynamics 3. Clasterization: trees in the spanning forest do not overlap.

27

Institute for Problems of Mechanical Engineering of RAS Control of Complex Systems Laboratory

Decentralized control of Lurie networks under bounded disturbances (Fradkov, Grigoriev, Selivanov, IEEE Conf. Dec.Contr. Orlando, Dec. 2011):

Control goal:

or i=1,…N

Institute for Problems of Mechanical Engineering of RAS Control of Complex Systems Laboratory

Decentralized control of Lurie networks under bounded disturbances (Fradkov, Grigoriev, Selivanov, IEEE Conf. Dec.Contr. Orlando, Dec. 2011):

Adaptive controller:

Lyapunov function: V(x1,…xN,t)=∑αi Qi(xi,t), αi >0 – some weights

29

Institute for Problems of Mechanical Engineering of RAS Control of Complex Systems Laboratory

Decentralized control of Lurie networks with delayed couplings (Fradkov, Selivanov, Fridman, 18th World Congress on Aut. Control, Milan, Aug. 2011):

Control goal:

30

Institute for Problems of Mechanical Engineering of RAS Control of Complex Systems Laboratory

Decentralized control of Lurie networks with delayed couplings (Fradkov, Selivanov, Fridman, 18th World Congress on Aut. Control, Milan, Aug. 2011):

Lyapunov functional:

Adaptive controller:

Phase and claster synchronization in a network of Landau-Stuart oscillators • Selivanov A.A., Lehnert J, Dahms T., Hovel P., Fradkov A.L., Schoell E. •Adaptive synchronization in delay-coupled networks of Stuart-Landau oscillators. Phys.Rev. E 85, 016201 (2012).

Institute of Problems in Mechanical Engineering of RAS Control of Complex Systems Laboratory

Phase and claster synchronization in a network of Landau-Stuart oscillators (cont)

Institute of Problems in Mechanical Engineering of RAS Control of Complex Systems Laboratory

Robot football at Math faculty of SPbSU

Institute of Problems in Mechanical Engineering of RAS Control of Complex Systems Laboratory

Robot football at Math faculty