analysis and design of robust and high-performance complex ...pietro/conferences/2017... · ms...

Analysis and Design of Robust and High-Performance Complex Dynamical Networks

Milad SiamiInstitute for Data, Systems, and Society (IDSS)

The 3rd Annual Meeting of SIAM Central States Section September 30 - October 1, 2017

Colorado State University, Fort Collins, Colorado

Performance

Fundamental Issues in Controlled Networks

How does the effect of network uncertainties propagate in large-scale dynamical networks?

M. Siami — SIAM2017

Performance

Fundamental Issues in Controlled Networks

How does the effect of network uncertainties propagate in large-scale dynamical networks?

Node Node

Server

Client

Energy Aware Control

Performance Measures for Power Networks:

Rate-limiting Systems / Distributed and Parallel Computations

Internet-scale applications presents a challenging technical problem.

Vehicle Formation Control:

Cloud-based Services:

The (extra) kinetic energy loss in the network in order to all vehicles reach a desired velocity with a desired formation pattern.

An Example of viable performance measures:

Phase Synchronization in Distributed Power Networks

Total resistive power loss due to returning a power network in Synchrony

An Example of performance measures:

Applications Areas

Part I: Performance Analysis and Tradeoffs

Part II: Network Synthesis for Performance Enhancement

● Growing Linear Consensus Networks

● Network Sparsification with Guaranteed Systemic Performance Measures

● Fundamental Limits and Tradeoffs in Linear Consensus Networks

● Centrality Measures in Linear Consensus Networks

● A New Notion: Systemic Performance Measures

First-order Consensus Networks

Each node is subject to disturbances

The model of a first-order consensus network:

A Simple ObservationHow does noise propagate in a consensus network?

N (L) :

�x(t) = �Lx(t) + �(t)

y(t) = Mx(t)

N (L) :

�x(t) = �Lx(t) + �(t)

y(t) = Mx(t)

How can we quantify this observation for arbitrary

networks?

- role of underlying graph- role of disturbances

• Steady-state expected dispersion

• norms

• Uncertainty volume of the output

• Entropy of the output

• Local deviation error

• Maximum spread in the output

• Steady-state expected value of any convex function of the output

• And many more

H2 H�

Existing Standard Measures to Quantify Noise Propagation

MS & N. Motee, “Fundamental Limits on Robustness Measures in Networks of Interconnected Systems,” CDC2013.MS & N. Motee, “Fundamental Limits and Tradeoffs on Disturbance Propagation in Large-Scale Dynamical Networks,” IEEE Transactions on Automatic Control, 2016B.

�H2(L) := lim

t��E

�y (t)y(t)

�Performance measure:

Theorem:

where the coupling graph is strongly connected and balanced.

A Typical Performance Measure

MS & N. Motee, “Fundamental Limits on Robustness Measures in Networks of Interconnected Systems,” CDC2013.MS & N. Motee, “Fundamental Limits and Tradeoffs on Disturbance Propagation in Large-Scale Dynamical Networks,” IEEE Transactions on Automatic Control, 2016B.

�H2(L) := lim

t��E

�y (t)y(t)

�Performance measure:

Theorem:

where the coupling graph is strongly connected and balanced.

A Typical Performance Measure

For connected undirected graphs, we get

Bamieh, M. Jovanovic, P. Mitra, and S. Patterson "Coherence in large-scale networks: Dimension-dependent limitations of local feedback," TAC2012.

10MS & N. Motee, “Fundamental Limits on Robustness Measures in Networks of Interconnected Systems,” CDC2013.MS & N. Motee, “Fundamental Limits and Tradeoffs on Disturbance Propagation in Large-Scale Dynamical Networks,” IEEE Transactions on Automatic Control, 2016

Minimal and Maximal Performance index

Universal Bounds and Scaling Laws

Graph-Dependent Fundamental Limits

MS & N. Motee, “Fundamental Limits and Tradeoffs on Disturbance Propagation in Large-Scale Dynamical Networks,” IEEE Transactions on Automatic Control, 2016

12MS & N. Motee, “Fundamental Limits and Tradeoffs on Disturbance Propagation in Large-Scale Dynamical Networks,” IEEE Transactions on Automatic Control, 2016

● A New Notion: Systemic Performance Measure

Other Sources of Uncertainty

Theorem:

�H2(L) := lim

t��E

�y (t)y(t)

Centrality Measures in Consensus Networks

M. Siami, S. Bolouki, B. Bamieh and N. Motee, “Centrality measures in linear consensus networks with structured network uncertainties”, IEEE Transaction on Control of Network Systems. Accepted.

Results:

l†ii L†

r{i,j} := l†ii + l†jj � 2l†ij�2

Centrality Measures in Consensus Networks

M. Siami, S. Bolouki, B. Bamieh and N. Motee, “Centrality measures in linear consensus networks with structured network uncertainties”, IEEE Transaction on Control of Network Systems. Accepted.

Example:

How Equal Are The Agents / Links? Uncertain Emitters

Uncertain Dynamics

Uncertain Sensors

Example:

Uncertain Dynamics

Uncertain Sensors

10Example:

Uncertain Dynamics

Uncertain Sensors

Example:

How Equal Are The Agents / Links?

● A New Notion: Systemic Performance Measure

• -norms

• Uncertainty volume of the output

• Entropy of the output

• Local deviation error

• Maximum spread in the output

• Steady-state expected value of any convex function of the output

• And many more

Other Performance Measures:

• Steady-state expected dispersion

Existing Standard Performance Measures

[Bamieh, Jovanovic]

Key Observation

Some of the closely related works in the literature:

(i) Monotonicity

(ii) Convexity

(iii) Orthogonal Invariance

A New Notion: Systemic Performance Measures

Cu,’

MS & N. Motee, "Performance Analysis of Linear Consensus Networks with Structured Stochastic Disturbance Inputs,” ACC2015. MS & N. Motee, "Schur-Convex Robustness Measures in Dynamical Networks, " ACC2014.

A New Notion: Systemic Performance Measures

The value of viable measure should be larger for sparser graphs.

A viable measure should be convex for network design purposes.

A viable measure should be orthogonal invariant.

Important Examples of Systemic Performance Measures

MS & N. Motee, “Growing Linear Consensus Networks via Systemic Performance Measures”, IEEE Transactions on Automatic Control, 2018, Accepted.

Several scenarios for improving the network performance or reducing network complexity:

(i) Rewiring (ii) Adding new links

(iv) Sparsification(iii) Optimum allocation of weights to a given network Topology

[Kolla 2006], [Spielman 2008],[Siami NecSys15]

[Siami ACC14][Siami CDC14]

[Boyd 2006], [Ghosh 2008],[Siami 2014], [Jovanovic, Fardad], …

[Ghosh 2006],[Summers, Dorfler],[Mesbahi, Zelazo],[Fardad CDC15],[Hassan-Moghadam ACC15], [Siami ACC16], …

Research Questions

Due to performance deterioration, agents are allowed to establish a few new communication links:

• Who should exchange information with whom?

• What are reasonable ways to compare various graph topology?

• How can we quantify the resulting performance improvement?

• Where should we add new links?

• By adding few new links, how much should we expect to improve the performance?

• What if we change the weights of candidate links?

Add three new linksAmong seven candidate links

#140 cases even for a small network

Example:

Adding New Links

Example:

Adding New Links

Example:

Adding New Links

minimizeE��k(Ec)

�(L + L)

�k(Ec) :=�E � Ec

�� |E | = k�

ϖ : Ec → R++

Ec = {e1, . . . , ep}

Our Optimization Problem

• This problem is combinatorial and nonconvex.

• A simpler version with is in fact NP-hard [Mosk-Aoyama, 2008]

ρ(L) = λ−12

Special Cases:

Our Results: Exact Solution for k = 1:

MS & N. Motee, “Tractable approximation algorithms for the NP-hard problem of growing linear consensus networks”, ACC2016.MS & N. Motee, “Growing Linear Consensus Networks via Systemic Performance Measures”, IEEE Transactions on Automatic Control, 2018, Accepted.

Theorem:

ρ (L) :=n!

ϕ(λi)

��k+2, . . . , �n, �, . . . , ��

�< minimize

E��k(Ec)�(L + L)

�(�i) < minimizeE��k(Ec)

�(L + L)

�lim�� (�) = 0

Examples: norm FOC norm SOC Zeta measure Gamma Entropy

H2 H�H2 H�

Quantify the Best Achievable Performance Bounds

Example: Adding One Link

A randomly selected coupling graph with 60 nodes and 176 links.

Label of added link

Example: Adding One Link

Link weight

Example: Hard Limits

Link weight

Example: Hard Limits

Hard limits on the best achievable performance of the network after adding k links to the network coupling graph.

Example: Hard Limits on The Best Achievable Performance

��k+2, . . . , �n, �, . . . , ��

�< minimize

E��k(Ec)�(L + L)

✔M. Siami — SIAM2017

��k+2, . . . , �n, �, . . . , ��

�< minimize

E��k(Ec)�(L + L)

✔M. Siami — SIAM2017

Two Approximation Methods:

•A Linearization-Based Approximation Method

•Simple Greedy by Sequentially Adding Links

Efficient Approximate Algorithms for k > 1:

Tr��(L)L

�(L + L)

�(L + �L) = �(L) + �Tr��(L)L

�+ O(�2).

Approximation Using Linearization

Corollary:

�(L + �L) = �(L) + �Tr��(L)L

�+ O(�2).

MS & N. Motee, “Tractable approximation algorithms for the NP-hard problem of growing linear consensus networks”, ACC2016.MS & N. Motee, “Growing Linear Consensus Networks via Systemic Performance Measures”, IEEE Transactions on Automatic Control, 2016, Submitted.

In general: The problem boils down to select the k-largest elements of the following set

Two Approximation Methods

•A Linearization-Based Approximation Method

•Simple Greedy by Sequentially Adding Links

Efficient Approximate Algorithms for k > 1:

Relaxation by Adding Links One at a Time

Complexity:

Relaxation by Adding Links One at a Time

Complexity:

Example: Comparing the Proposed Methods

�(L + L)k: number of links we want to add

: red dashed links minimizeE��k(Ec)

�(L + L)

Computational Complexity

The Greedy Method:

The Brute-Force Method:

The Linearization Method:

Computational Complexity

The Greedy Method:

The Brute-Force Method:

The Linearization Method:

Improving Performance & Robustness

Fundamental Limits and Tradeoffs

Network Structures

Network Design Strategies

Meaningful Measures• Spectral Systemic Measure• Systemic Performance Measure• Centrality Measure• Nodal Performance Measure

Network Structures

Meaningful Measures

• Spectral Systemic Measure• Systemic Performance Measure• Centrality Measure• Nodal Performance Measure

Network Structures

Concluding Remarks

• Propose tractable approximate methods with computable/guaranteed performance bounds

• Quantify the best achievable performance bounds for the network synthesis problem

• Developing a graph-theoretic framework to relate the underlying structure of the system to its overall performance measure

• Characterize fundamental limits on robustness and performance measures of Nonlinear autocatalytic pathways

• Introducing new insights into the network centrality based not only on the network graph but also on a more structured model of network uncertainties.

• Characterize inherent tradeoffs between various systemic measures

4623/24

Journal Papers:1. M. Siami and N. Motee, “New Bounds on H2-Norm of Noisy Linear Dynamical Networks”, Automatica. 2017.

2. M. Siami and N. Motee, “Fundamental Limits and Tradeoffs on Disturbance Propagation in Large-Scale Dynamical

Networks”, IEEE Transactions on Automatic Control, 2016.

3. M. Siami, and J. Skaf, “Structural Analysis and Optimal Design of Distributed System Throttlers”, IEEE Transactions on

Automatic Control, 2017.

4. M. Siami, and N. Motee, “Growing Linear Consensus Networks via Systemic Performance Measures”, IEEE Transactions

on Automatic Control, Accepted.

5. M. Siami, S. Bolouki, B. Bamieh and N. Motee, “Centrality measures in linear consensus networks with structured network

uncertainties”, IEEE Transaction on Control of Network Systems. 2017.

6. M. Siami, and N. Motee, “Abstraction of Linear Consensus Networks with Guaranteed Systemic Performance Measures”,

IEEE Transactions on Automatic Control, 2016, Submitted.

References:

My side projects at Lehigh

• Prof. R. P. Malhame, Polytechnique Montréal (Opinion Dynamics)• Dr. C. Somarakis, Lehigh U (Systemic Risk Measures) • Mr. Y. Ghaedsharaf, Lehigh U (Time-delay)

Collaborators During My Ph.D. Study

• Prof. N. Motee, Lehigh U (My Advisor)• Prof. B. Bamieh, UCSB • Prof. M. Khammash, ETH• Prof. J. C. Doyle, Caltech• Prof. G. Buzi, ETH• Dr. S. Bolouki, UIUC• Dr. J. Skaf, Google

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