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Analysis and Design of Robust and High-Performance Complex Dynamical Networks
Milad SiamiInstitute for Data, Systems, and Society (IDSS)
MIT
The 3rd Annual Meeting of SIAM Central States Section September 30 - October 1, 2017
Colorado State University, Fort Collins, Colorado
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Performance
Fundamental Issues in Controlled Networks
How does the effect of network uncertainties propagate in large-scale dynamical networks?
M. Siami — SIAM2017
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Performance
Fundamental Issues in Controlled Networks
How does the effect of network uncertainties propagate in large-scale dynamical networks?
Node Node
Node Node
Node Node
Server
Node
Client
Client
Client
M. Siami — SIAM2017
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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
M. Siami — SIAM2017
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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
M. Siami — SIAM2017
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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
M. Siami — SIAM2017
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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
M. Siami — SIAM2017
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First-order Consensus Networks
Each node is subject to disturbances
The model of a first-order consensus network:
M. Siami — SIAM2017
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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
M. Siami — SIAM2017
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• 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
M. Siami — SIAM2017
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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
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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.
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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
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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
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12MS & N. Motee, “Fundamental Limits and Tradeoffs on Disturbance Propagation in Large-Scale Dynamical Networks,” IEEE Transactions on Automatic Control, 2016
Graph-Dependent Fundamental Limits
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12MS & N. Motee, “Fundamental Limits and Tradeoffs on Disturbance Propagation in Large-Scale Dynamical Networks,” IEEE Transactions on Automatic Control, 2016
Graph-Dependent Fundamental Limits
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● Fundamental Limits and Tradeoffs in Linear Consensus Networks
● Centrality Measures in Linear Consensus Networks
● A New Notion: Systemic Performance Measure
Part I: Performance Analysis and Tradeoffs
M. Siami — SIAM2017
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Other Sources of Uncertainty
M. Siami — SIAM2017
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Other Sources of Uncertainty
M. Siami — SIAM2017
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Other Sources of Uncertainty
M. Siami — SIAM2017
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Other Sources of Uncertainty
M. Siami — SIAM2017
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Other Sources of Uncertainty
M. Siami — SIAM2017
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Other Sources of Uncertainty
M. Siami — SIAM2017
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Other Sources of Uncertainty
M. Siami — SIAM2017
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Other Sources of Uncertainty
M. Siami — SIAM2017
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Other Sources of Uncertainty
M. Siami — SIAM2017
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Other Sources of Uncertainty
M. Siami — SIAM2017
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Other Sources of Uncertainty
M. Siami — SIAM2017
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Other Sources of Uncertainty
M. Siami — SIAM2017
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Theorem:
�H2(L) := lim
t��E
�y (t)y(t)
�
where
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.
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Results:
di
l†ii L†
r{i,j} := l†ii + l†jj � 2l†ij�2
i
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.
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Example:
How Equal Are The Agents / Links? Uncertain Emitters
Uncertain Dynamics
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How Equal Are The Agents / Links? Uncertain Emitters
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How Equal Are The Agents / Links? Uncertain Emitters
Uncertain Dynamics
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Example:
How Equal Are The Agents / Links?
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● Fundamental Limits and Tradeoffs in Linear Consensus Networks
● Centrality Measures in Linear Consensus Networks
● A New Notion: Systemic Performance Measure
Part I: Performance Analysis and Tradeoffs
M. Siami — SIAM2017
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• -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
✔
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[Bamieh, Jovanovic]
Key Observation
Some of the closely related works in the literature:
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(i) Monotonicity
(ii) Convexity
(iii) Orthogonal Invariance
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A New Notion: Systemic Performance Measures
M. Siami — SIAM2017
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Cu,’
Cu,’
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.
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Important Examples of Systemic Performance Measures
MS & N. Motee, “Growing Linear Consensus Networks via Systemic Performance Measures”, IEEE Transactions on Automatic Control, 2018, Accepted.
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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], …
Part II: Network Synthesis for Performance Enhancement
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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], …
Part II: Network Synthesis for Performance Enhancement
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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], …
Part II: Network Synthesis for Performance Enhancement
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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], …
Part II: Network Synthesis for Performance Enhancement
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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], …
Part II: Network Synthesis for Performance Enhancement
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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], …
Part II: Network Synthesis for Performance Enhancement
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26
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], …
Part II: Network Synthesis for Performance Enhancement
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26
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], …
Part II: Network Synthesis for Performance Enhancement
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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?
M. Siami — SIAM2017
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• 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
M. Siami — SIAM2017
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• 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
M. Siami — SIAM2017
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• 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
M. Siami — SIAM2017
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minimizeE��k(Ec)
�(L + L)
�k(Ec) :=�E � Ec
�� |E | = k�
L E
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
M. Siami — SIAM2017
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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.
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Theorem:
ρ (L) :=n!
i=2
ϕ(λi)
���k+2, . . . , �n, �, . . . , �� �� �
k
�< minimize
E��k(Ec)�(L + L)
n�
i=k+2
�(�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
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.
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Example: Adding One Link
A randomly selected coupling graph with 60 nodes and 176 links.
M. Siami — SIAM2017
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Label of added link
Perfo
rman
ce M
easu
re
Example: Adding One Link
M. Siami — SIAM2017
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Link weight
Perfo
rman
ce M
easu
re
Example: Hard Limits
M. Siami — SIAM2017
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Link weight
Perfo
rman
ce M
easu
re
Example: Hard Limits
M. Siami — SIAM2017
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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
M. Siami — SIAM2017
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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, �, . . . , �� �� �
k
�< minimize
E��k(Ec)�(L + L)
✔M. Siami — SIAM2017
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35
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, �, . . . , �� �� �
k
�< minimize
E��k(Ec)�(L + L)
✔M. Siami — SIAM2017
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Two Approximation Methods:
•A Linearization-Based Approximation Method
•Simple Greedy by Sequentially Adding Links
Efficient Approximate Algorithms for k > 1:
M. Siami — SIAM2017
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minimizeE��k(Ec)
Tr���(L)L
�
minimizeE��k(Ec)
�(L + L)
�(L + �L) = �(L) + �Tr���(L)L
�+ O(�2).
Approximation Using Linearization
M. Siami — SIAM2017
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Corollary:
�(L + �L) = �(L) + �Tr���(L)L
�+ O(�2).
Approximation Using Linearization
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
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In general: The problem boils down to select the k-largest elements of the following set
Approximation Using Linearization
M. Siami — SIAM2017
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Two Approximation Methods
•A Linearization-Based Approximation Method
•Simple Greedy by Sequentially Adding Links
Efficient Approximate Algorithms for k > 1:
M. Siami — SIAM2017
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Relaxation by Adding Links One at a Time
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.
Complexity:
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Relaxation by Adding Links One at a Time
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.
Complexity:
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Example: Comparing the Proposed Methods
minimizeE��k(Ec)
�(L + L)k: number of links we want to add
: red dashed links minimizeE��k(Ec)
�(L + L)
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Computational Complexity
The Greedy Method:
The Brute-Force Method:
The Linearization Method:
MS & N. Motee, “Growing Linear Consensus Networks via Systemic Performance Measures”, IEEE Transactions on Automatic Control, 2018, Accepted.
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Computational Complexity
The Greedy Method:
The Brute-Force Method:
The Linearization Method:
MS & N. Motee, “Growing Linear Consensus Networks via Systemic Performance Measures”, IEEE Transactions on Automatic Control, 2018, Accepted.
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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
M. Siami — SIAM2017
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Improving Performance & Robustness
Fundamental Limits and Tradeoffs
Network Structures
Network Design Strategies
I I
I
Meaningful Measures• Spectral Systemic Measure• Systemic Performance Measure• Centrality Measure• Nodal Performance Measure
M. Siami — SIAM2017
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Improving Performance & Robustness
Fundamental Limits and Tradeoffs
Network Structures
Network Design Strategies
I I
III
IIII
Meaningful Measures• Spectral Systemic Measure• Systemic Performance Measure• Centrality Measure• Nodal Performance Measure
M. Siami — SIAM2017
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Meaningful Measures
Network Design Strategies
• Spectral Systemic Measure• Systemic Performance Measure• Centrality Measure• Nodal Performance Measure
Fundamental Limits and Tradeoffs
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
M. Siami — SIAM2017
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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:
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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