joint spectral radius of rank one matrices and the maximum...
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Joint Spectral Radius of Rank One Matrices and the Maximum Cycle Mean Problem
Amir Ali Ahmadi Pablo A. Parrilo MITIBM MIT
IEEE Conference on Decision and Control Maui, Hawaii
December 10, 2012
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The Joint Spectral Radius
Given a finite set of matrices
Joint spectral radius (JSR):
If only one matrix:
Spectral Radius
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JSR and switched linear systems
Joint spectral radius (JSR):
“Stable under arbitrary switching” iff
Switched linear dynamics:
Linear dynamics:
“Stable” iff
Spectral radius:
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Undecidability and the rational finiteness conjecture
[Blondel, Tsitsiklis]
If only one matrix:
For more than one matrix:
(even for 2 matrices of size 47x47 !!) [Blondel, Canterini]
Open problem: decidability of testing
Would become decidable if rational finiteness conjecture is true
Fact: (spectral radius of any matrix product of length k)1/k ≤JSR Finiteness property: equality achieved at finite k
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Computation of the JSR Applications
Stability of switched systems, capacity of codes, asymptotics of overlap-free words, convergence of consensus algorithms, continuity of wavelet functions, trackability of graphs, …
Methods of approximation
CQLF [Ando, Shih], Kronecker liftings [Blondel, Nesterov], SOS [Parrilo, Jadbabaie], Path dependent Lyapunov functions [Lee, Dullerud], Polytopic norms [Guglielmi, Protasov, Zennaro], Path-complete graph Lyapunov functions [AAA, Jungers, Parrilo, Roozbehani], …
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Focus of this 2-page paper: JSR of rank one matrices
An O(m3+m2n) algorithm for computing the exact JSR
Based on a connection to a problem in graph theory
As a corollary: finiteness property & some structural property of the optimal word
Motivated by recent work of [Liu, Xiao `11]
Correction to a claim in their paper
But our main contribution is:
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Basic facts about rank one matrices
A rank one iff
spectral radius:
Products of rank one matrices have rank at most one:
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Cycles and cycle gains
Easy definitions: • Cycle • Simple cycle • Cycle gain
• Maximum cycle gain
• Gain-maximizing cycle
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From matrix products to cycles in graphs
Nodes: matrices
Complete directed graph on m nodes:
Edge weights:
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Maximum cycle gain gives the JSR
Thm: Let cmax be a gain-maximizing cycle, with lmax and ρcmax denoting its length and the product of the weights on its edges, respectively. Then, the joint spectral radius is given by:
Proof sketch:
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Finiteness property and the optimal product
Corollary:
• The JSR is achieved by the spectral radius of a finite matrix product, of length at most m. (In particular, the finiteness property holds.)
• There always exists an optimal product where no matrix appears more than once.
Proof: A simple cycle does not visit a node twice.
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Not fun to enumerate all simple cycles…
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Maximum Cycle Mean Problem (MCMP)
• Cycle mean
• Maximum cycle mean (MCM)
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Karp’s algorithm for MCMP
Running time O(|N||E|)
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Take logs and apply Karp
Nodes: matrices
Complete directed graph on m nodes:
Edge weights:
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2 remarks, 2 examples, and that’s it…
(1)
There is a different and independent characterization of the JSR of rank one matrices due to Gurvits and Samorodnitsky (2005), brought to our attention by Vincent Blondel
Leads to solving a geometric program
(2)
Our approach based on MCMP can directly be used for deciding stability of rank one matrices under constrained switching
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Length m may indeed be needed
Counterexample to the claim by [Liu, Xiao ‘11] (claim was that only products of length at most 2 are needed)
Spectral radius of all products is zero except for cyclic repetitions of
Hence
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Common quadratic Lyapunov function can fail
(can be proven e.g. using our algorithm)
An LMI searching for a common quadratic Lyapunov function can only prove
In fact, CQLF does “as badly as possible” on this example
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Thank you for your attention!
Questions?
Want to know more?
http://aaa.lids.mit.edu/