dynamical networks: structural analysis and...
TRANSCRIPT
Giulia GiordanoDipartimento di Matematica e Informatica
Università degli Studi di Udine
DYNAMICAL NETWORKS:STRUCTURAL ANALYSIS
AND SYNTHESIS
March 31st, 2015
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DYNAMICALNETWORKS
telecommunication / data communication networks
production and distribution systems
(bio)chemical processes
A B C
water distribution networks
synchronizationtraffic management
social networks
biological systemsand ecosystems
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telecommunication / data communication networks
production and distribution systems
(bio)chemical processes
A B C
water distribution networks
synchronizationtraffic management
social networks
biological systemsand ecosystems
1 2
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1 2
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Dynamical network:subsystems (each with its own dynamics)interconntected by "flows" / interactions
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(hyper-)graph representation:
nodes→subsystemsarcs→flows
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1 2
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hyper-graph:flows can require
interactions among more than two compartments
• positive systems• “mass” conservation constraints• flow: interaction between two compartments
graph representation:
nodes→compartmentsarcs→flows
1 2
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Compartmental systems...
...generalised
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Giulia GiordanoA twofold goal
Structural Analysis
...especially meant for...
Structural Control Synthesis
- assess properties independent of parameter values
- based only on the nature of interactions among subsystems (graph topology)
- explain robustness and characteristic behaviours
- interactions not spontaneous, decided based on local information (decentralized)
- control / coordinate the global system behaviour
Natural Systems Man-made Systems
biologically-inspired design: learn from nature
synthetic biology:engineer natural systems
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Structural Analysis
Structural Control Synthesis
→Applications to biochemical systems
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explain behavioursbased on the systeminherent structure
Biological systems are extremely robust:fundamental properties are always preserved
despite huge uncertainties and parameter variations
structure graph
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STRUCTURAL ANALYSIS
(...motifs...)
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Structural: more than robust.
Structural property:satisfied by all the systems of a family
specified by a structure without numerical bounds
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Structural investigation: properties of interest
?
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Structural investigation: system specifications
!
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Piecewise-linear Lyapunov functionsfor structural stability
of biochemical networks
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F. Blanchini, G. Giordano, Automatica, vol. 50, n. 10, 2014
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polyhedral (piecewise-linear) Lyapunov functions
x1
x2
x3
x4
X = [ x1 x2 x3 x4 ]
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ODEsystem:
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g0 =influx
ODE system:
S g
Graph:
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polyhedral (piecewise-linear) Lyapunov function?
nonlinear model
monotonedissipativity:
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...lineardifferentialinclusion
diagonal nonnegative matrix
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Absorb the system in a...
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Example:
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Their robust stability is equivalent→ same (weak) polyhedral Lyapunov function
Linear differential inclusionassociated with a
linear discrete difference inclusion
Numerical procedurefor finding the unit ball
of the polyhedral Lyapunov function(if any)
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converges to
The system is structurally stable!
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Example:
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To prove asymptotic stability→introduce infinitesimal dissipativity
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Enzymatic reaction
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Metabolic network
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Boundedness
• Absorb the system in a positive linear differential inclusion
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stability boundedness
Yes
No
Yes
No
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Structural graph-based classificationof oscillatory and multistationarybehaviours in biochemical systems
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F. Blanchini, E. Franco, G. Giordano, Bull Math Biol, vol. 76, n. 10, 2014
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The only admissible instability is oscillatory
All the cycles in the Jacobian graph are negative...
General classification for sign definite-systems:
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Candidate oscillator
Candidate multistable
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Interconnection of monotone subsystems → similar results
Negative feedback interconnectionof two structurally stable monotone subsystems:
oscillator
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Structural steady-state analysis
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Input variation
Output variation(at steady state)
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Increase Decrease
Perfect adaptation
Depends on parameters
Efficient numerical algorithmbased on J = BDC decomposition
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Influence matrix:steady-state effectof each variable
on the others
Network fromShinar&Feinberg (2010)
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Influence matrix:steady-state effectof each variable
on the others
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Structural Analysis
Structural Control Synthesis
→Network-decentralized control
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huge number of subsystems
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Control strategy decided locally,but the goal is to govern
the dynamics of the whole system
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Decentralized control: the feedback matrix has a structure given by the graph
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We can find a control that is
And under control saturation?
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Acknowledgements:
joint work with
Christian Cuba Samaniego, Elisa Franco,Vahid Mardanlou, Pier Luca Montessoro
my supervisor Franco Blanchini,
VIELEN DANK!
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SOMEEXTRASLIDES
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lineardifferenceinclusion
(1)
(2)
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the trajectoriesof (1)
point inside......the cone
defined by the trajectories of (2)
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(1) robustly stable iff (2) robustly stablewith the same (weak) polyhedral Lyapunov function
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Numerical procedure
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local vs global asymptotic stability
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• A sequence of sets is computedwhich evolves in the positive orthant
• The same idea of the previous procedurecan be applied starting from the unit simplex
BoundednessGiulia Giordano