Computing over Unreliable Computing over Unreliable C C - PowerPoint PPT Presentation

Computing over Unreliable Computing over Unreliable C C Communication Networks Communication Networks i i i i N N k k Nicola Elia Joint work with Jing Wang D Dept. of Electrical and Computer Engineering t f El t i l d C t E i i Iowa State University Acknowledgments to: NSF

Interconnected systems Interconnected systems Interconnected systems Interconnected systems Materials 3 2 System Biology y gy P P A P P A Large dimensions Computers networks Many nonlinearities Power grid P A P A 1 4 Uncertainty in the interactions Avionics systems Lots of feedback loops Lots of feedback loops Economics & Finance P A P A Not clear separations Ecology 5 6 Traffic Social Networks Social Networks Multi-agents systems Difficult to analyze/design, abrupt changes, complex unpredicted behaviors What are the determining factors?

Interconnected systems: New opportunities Interconnected systems: New opportunities Interconnected systems: New opportunities Interconnected systems: New opportunities New applications which are network distributed Estimation 3 2 Detection P A P A Control O ti Optimization i ti P A P P P A 1 4 Computation P A P A 5 5 6 6 New developments Ne de elopments Integrated theory of control and information Dynamical system view of distributed computing algorithms y y p g g Focus on multi-agent systems with “simple” agents

Channels in the Loops Channels in the Loops Channels in the Loops Channels in the Loops How do communication channels affect networked systems? Concentrate on channel “fading” and additive noise Concentrate on channel fading and additive noise P A P A A A Uncertainty in the interactions P A P A Lots of feedback loops P A P A

Protocol Design Protocol Design Protocol Design Protocol Design (Elia Eisenbeis TAC11 Padmasola Elia 06) (Elia, Eisenbeis TAC11, Padmasola Elia 06) New protocols need to focus on data freshness rather than data integrity ξ 1 u 1 y p P z -1 ACK ξ 2 ξ 1 ξ 1 ξ s QoS for MS stability y u 1 ξ 2 y p Controller QoS % ACK lost % ACK lost Actuator-Sensor 70% packet drop, Service channel 50% ACK losses

Outline Outline Outline Outline Unreliable Networks (Fading Network Framework) Networked control approach to distributed computation of averages averages Limitations due to unreliable communication Emergence of complex behavior Mitigation techniques New perspective on distributed optimization systems Distributed optimization over unreliable networks Distributed optimization over unreliable networks

Fading Channels as Uncertain Systems Fading Channels as Uncertain Systems Fading Channels as Uncertain Systems Fading Channels as Uncertain Systems Intermittent channel with probability e μ Δ Re-prarametrization(s) Model for packet loss in networks (concentrate on fading neglect quantization) Special case of analog memory-less multiplicative channel Extends to Gaussian fading channels also with memory

A Simple Problem A Simple Problem A Simple Problem A Simple Problem LTI x Plant Plant x ( k ) state (r.v.) of the system at time k k ( k ) t t ( ) f th t t ti e Q ( k )= E { x ( k ) x ( k )’ } K K Linear gain Mean Square Stability Mean Square Stability Noiseless With With white-noise input hit i i t Minimal Channel Quality for Mean Square Stability?

A general framework: the Fading Network A general framework: the Fading Network A general framework: the Fading Network A general framework: the Fading Network (Elia 05) (Elia 05) Fading Network = Mean Network + Uncertainty Stochastic St h ti w z Uncertainty is Stochastic y p u IID in k , Independent in i N P K Zero Mean, var = u p y n G M Mean Network, N , deterministic LTI M deterministic LTI MS Stability margin

MS Stability Robustness Analysis MS Stability Robustness Analysis MS Stability Robustness Analysis MS Stability Robustness Analysis (Elia 05) (Elia 05) Given stable with strictly upper/lower triangular Let w z y p u P K N u p y Then n G M spectral radius Separation result CL system MS stable iff Based on ElGaoui 95, Ku Athans 77, Willems Blankenship 71, Kleinman 69 Wonham 67. Related to El Bouhtouri et all 02, Jianbo Lu Skelton 02.

State State-Feedback with One Channel State State Feedback with One Channel Feedback with One Channel Feedback with One Channel State feedback with one memoryless multiplicative channel at the plant input the plant input x r w Plant Plant Plant Δ ξ ξ z μ K K K For the intermittent channel :

Why is single loop stabilization relevant? Why is single loop stabilization relevant? Why is single loop stabilization relevant? Why is single loop stabilization relevant? 3 2 1 - e Collection of interconnected stable system can be unstable P A P A P A P A 1 4 P A P A A A S System t 5 6 • Stable if link 6-1 is present • Unstable if link 6-1 is absent 3 2 e e • Mean Stable is e < 0.517 P A P A P A P A • Mean Square stable if e < 0.501 P A P A 1 4 system system P A P A 5 6 Link 6-> 1 Similar to one fading channel in the loop

Limitations for Multi Limitations for Multi agent Systems. Limitations for Multi-agent Systems. Limitations for Multi agent Systems. agent Systems. 3 2 w z P A P A y p P A P A P A 1 4 4 N N A A P A u p P A P A n M M 5 5 6 6 Many channels in many loops Same tool applies QoS analysis more complex Simple mechanism for emergence of complex behavior Simple mechanism for emergence of complex behavior

Consensus: a paradigm for distributed computation Consensus: a paradigm for distributed computation Consensus: a paradigm for distributed computation Consensus: a paradigm for distributed computation x ( 0 ) Discrete-time 3 2 P A P A P A 0 u u x x P A P A P A 1 4 Continuous-time P A P A P A P A L L 5 6 All the nodes are the same. Each node use the relative error from its neighbors to update its own state. The neighbors are determined by a graph. Under certain conditions Tsitsiklis, Olfati-Saber,Scutari, Fax, Murray, Zampieri, Fagnani, Cortes, Pesenti, Giulietti, Ren, Beard, Papachristodoulou, Lee, Jadbabaie, Low,….

Basic Graph Basic Graph Theory Basic Graph Basic Graph Theory Theory Theory ( Laplacian Matrix ) We can associate each edge with a positive weight , the Laplacian matrix is defined as Example: For 0-1 weights 1 2 4 3 L 1 =0 The left eigenvector of L associated with zero eigenvalue is all positive if the graph is strongly connected ( Balanced Laplacian ) ( Balanced Laplacian ) satisfy , needed for averaging satisfy , needed for averaging

Limitations on Information Exchange Limitations on Information Exchange Limitations on Information Exchange Limitations on Information Exchange Averaging over unreliable channels + noise ? β β update gain >= 0 d t i 0 ξ ij packet drop ij channel Pr ( ξ ij ( k ) = 1 ) = μ ij = QoS v i total additive noise to node i ; N(0,1) The model describes very simple-minded interacting agents Assume μ ij = μ for simplicity

Fading Network and system decomposition Fading Network and system decomposition Fading Network and system decomposition Fading Network and system decomposition Uncertainty re-parametrization State-space Equations for ( M , ∆ ) w z y p p M ( A B C ) has special structure M =( A , B , C ) has special structure P P A N P A u p v M

System Decomposition: Block Diagram System Decomposition: Block Diagram System Decomposition: Block Diagram System Decomposition: Block Diagram Conserved z FB C w System M P A N P A z -1 Deviation PB PB C C n n M System PAP n n z w ∆ Decomposition : Conserved + Deviation state When there is no noise or fading, is the consensus value, goes to zero

Emergence of new collective complex behavior Emergence of new collective complex behavior Emergence of new collective complex behavior Emergence of new collective complex behavior [Wang Elia MTNS10, TAC12] 1 2 6 3 5 4 Agents‘ states Loglog plot of the increments of x 1 Moment instability leads to power laws behaviors (under suitable assumptions) Integration of process with unbounded second moment (Levi’s processes)

Emergence of new collective complex behavior Emergence of new collective complex behavior Emergence of new collective complex behavior Emergence of new collective complex behavior For directed IID switching and strongly connected mean graph, assume the deviation system converges to an invariant distribution driven by Gaussian noise. Then is a hyper-jump-diffusion Then i h j diff i Deviation system is Mean Square unstable Deviation system is Mean Square unstable is a Levy flight, for a two-node system ([Kesten]) Emergent complex behavior is global (collective) Emergent complex behavior is global (collective) Long range impact of local criticality.

Levy flights vs. Normal random walk Levy flights vs. Normal random walk Levy flights vs. Normal random walk Levy flights vs. Normal random walk Two agent Levy flight Normal random walk β =1.35, μ =0.5, σ 2 =0.02, In the distribution of human travel [Brockmann] In economics and financial series [Mandelbrot, Sornette, Mantegna] In foraging search patterns of several species [Raynolds, Bartumeus] Exploitation cooperative searches and optimization? Exploitation cooperative searches and optimization? Mitigation strategies ?