Hierarchical Performance Analysis of Uncertain Large Scale ... · Synchronization speciﬁcations...

Hierarchical Performance Analysis of UncertainLarge Scale Systems

K. LaibA. Korniienko G. Scorletti F. Morel

Laboratoire AmpèreÉcole Centrale de Lyon

GT MOSAR, Grenoble, 6 October, 2015

K. Laib et al. (ECL) Hierarchical Robustness Analysis 6 Oct 2015 1 / 28

1 IntroductionMotivationProblem formulationProblem analysis

2 Proposed approachRobustness analysis and QC PropagationHierarchical approach

3 Application Example

4 Discussion

5 Conclusion and future work


Introduction Motivation

Context : PLL network

Large Scale Systems (LSS) : Phase Locked Loop (PLL) network

PLL network to deliver clock signal to synchronous multi-core processors

How to guarantee synchronization ?







4 3 2 1

8 7 6 5

12 11 10 9

16 15 14 13

+

-

Introduce global synchronization error







4 3 2 1

8 7 6 5

12 11 10 9

16 15 14 13

+

-

Introduce global synchronization error

Synchronization specifications (performance) are guaranteed ifTrg−→eg satisfies some frequency constraints



Context : Performance

Performance is expressed in frequency domain.

100

101

102

103

104

105

-80

-70

-60

-50

-40

-30

-20

-10

0

10

20

30

frequency, rad/sec

Mag

nit

ud

e, d

B

S1

ω0

+40 dB/dec

Max peak





100

101

102

103

104

105

-80

-70

-60

-50

-40

-30

-20

-10

0

10

20

30

frequency, rad/sec

Mag

nit

ud

e, d

B

S2

S1

ω0

+40 dB/dec

Max peak





100

101

102

103

104

105

-80

-70

-60

-50

-40

-30

-20

-10

0

10

20

30

frequency, rad/sec

Mag

nit

ud

e, d

B

S2

S1

ω0

+40 dB/dec

Max peak

0 0.05 0.1 0.15 0.2 0.25-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

time, sec

ε 1(t

), u

.e.

S1

S2



Context : Uncertainties

Active clock distribution network

Technological dispersions, modeling errors =⇒ uncertainties (∆)





Technological dispersions, modeling errors =⇒ uncertainties (∆)Uncertain subsystems






Uncertain Network






Uncertain Network

Robustness analysis :Perform the worst case robustness analysis for all the uncertainties ∆i




100

101

102

103

104

105

106

-140

-120

-100

-80

-60

-40

-20

0

20

frequency, rad/sec

Mag

nit

ud

e, d

B

nominal




100

101

102

103

104

105

106

-140

-120

-100

-80

-60

-40

-20

0

20

frequency, rad/sec

Mag

nit

ud

e, d

B

nominal

with uncertainties

upper bound

+40 dB/dec

ω0

Max peak

Synchronization specifications (performance) are guaranteed if the upper boundsatisfies the frequency constraints


Introduction Problem formulation

PLL network Performance

16 PLLs mutually synchronized

4 3 2 1

8 7 6 5

12 11 10 9

16 15 14 13

+

-

Two uncertain parameters for every PLL =⇒ 32 uncertain parameters

Nowadays networks : 100 PLLs =⇒ 200 uncertain parameters=⇒ classic method is not applicable

16 PLL network to show classic method results

Objective Compute an upper bound on ‖Trg→eg‖ for all the uncertainties


Introduction Problem analysis

Problem analysis

Large scale robustness analysis : two aspects problem

1 Robustness analysis : IQC based analysis (input-output description)

2 Large scale : decomposition techniques from graph theory



Problem analysis




Direct application of IQC based analysis =⇒ important computation time



Problem analysis





Few methods combining the two aspects : [Andersen et al., 2014]

Complex large scale

analysis problem

Complex large scale

optimization problem

Small scale

optimization problems

Modeling Optimization

High complexity

Low complexity

Specific solver

Large scale IQC analysis

Decomposition techniques



Problem analysis





Few methods combining the two aspects : [Andersen et al., 2014]

Complex large scale

analysis problem

Complex large scale


Small scale


Simplified small scale

analysis problems

(distributed)


High complexity

Low complexity

Specific solver



Hierarchical techniques

Small scale IQC analysis Standard

solver


Proposed approach Robustness analysis and QC Propagation

Integral Quadratic Constraints (IQC)

Integral Quadratic Constraints (IQC)

∫ +∞

−∞

(z(jω)w(jω)

)∗ΦP(jω)

(z(jω)w(jω)

)dω ≥ 0

Possibility to cover classical characterizations of performance

L2 gain∫ +∞

0

‖z(t)‖2 dt ≤ γ2

∫ +∞

0

‖w(t)‖2 dt⇐⇒∫ +∞

−∞

(z(jω)w(jω)

)∗ (−I 00 γ2

)(z(jω)w(jω)

)dω ≥ 0

Passivity∫ +∞

0

z(t)T w(t)dt ≥ 0⇐⇒∫ +∞

−∞

(z(jω)w(jω)

)∗ (0 II 0

)(z(jω)w(jω)

)dω ≥ 0



Proposed approach

Linear Time Invariant Systems

z(jω) = T(jω)w(jω) and QC based analysis

Frequency domain : frequency response at ω0

Performance : compute an upper bound on the frequency response (σ(T) ≤ γ)min

γγ s.t.

(TI

)∗ (−I 00 γ2I

)(TI

)≥ 0

100

101

102

103

104

105

106

-140

-120

-100

-80

-60

-40

-20

0

20

frequency, rad/sec

Mag

nit

ud

e, d

B

nominal

with uncertainties

upper bound

+40 dB/dec

ω0

Max peak

General performance(

TI

)∗ ( X YY∗ Z

)(TI

)≥ 0



Proposed approach : Robust Performance Theorem (LTI systems)

QC for performance and uncertainty : Classical interpretation

Theorem (Robust Performance Theorem)

T is X, Y, Z dissipative i.e.(TI

)∗ ( X YY∗ Z

)(TI

)≥ 0 ∀ ∆ ∈ ∆ =⇒ QC of T

if and only if

1)(

∆I

)∗ (Φ11 Φ12Φ∗12 Φ22

)(∆I

)≥ 0 ∀∆ ∈ ∆ =⇒ QC of ∆

2)(

MI

)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0

0 Y∗ 0 Z

(MI

)> 0







)∗ ( X YY∗ Z

)(TI

)≥ 0 ∀ ∆ ∈ ∆ =⇒ QC of T

if and only if

1)(

∆I

)∗ (Φ11 Φ12Φ∗12 Φ22

)(∆I

)≥ 0 ∀∆ ∈ ∆ =⇒ QC of ∆

2)(

MI

)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0

0 Y∗ 0 Z

(MI

)> 0

Condition 1) : infinite dimensional







)∗ ( X YY∗ Z

)(TI

)≥ 0 ∀ ∆ ∈ ∆ =⇒ QC of T

if and only if

1)(

∆I

)∗ (Φ11 Φ12Φ∗12 Φ22

)(∆I

)≥ 0 ∀∆ ∈ ∆ =⇒ QC of ∆

2)(

MI

)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0

0 Y∗ 0 Z

(MI

)> 0


Parametrize Φ with Φ∆ in 1) and test 2) =⇒ Construct a ’basis’ Φ∆ for Φ







)∗ ( X YY∗ Z

)(TI

)≥ 0 ∀ ∆ ∈ ∆ =⇒ QC of T

if and only if =⇒ if ( and only if )1) ∃ Φ ∈ Φ∆ =⇒ QC of ∆

2)(

MI

)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0

0 Y∗ 0 Z

(MI

)> 0


Parametrize Φ with Φ∆ in 1) and test 2) =⇒ Construct a ’basis’ Φ∆ for Φ







)∗ ( X YY∗ Z

)(TI

)≥ 0 ∀ ∆ ∈ ∆ =⇒ QC of T


2)(

MI

)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0

0 Y∗ 0 Z

(MI

)> 0


Parametrize Φ with Φ∆ in 1) and test 2) =⇒ Construct a ’basis’ Φ∆ for Φ=⇒ conservative (pessimist) results







)∗ ( X YY∗ Z

)(TI

)≥ 0 ∀ ∆ ∈ ∆ =⇒ QC of T


2)(

MI

)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0

0 Y∗ 0 Z

(MI

)> 0


Parametrize Φ with Φ∆ in 1) and test 2) =⇒ Construct a ’basis’ Φ∆ for Φ=⇒ conservative (pessimist) results

Conservatism depends on Φ∆




QC for performance and uncertainty : New interpretation



)∗ ( X YY∗ Z

)(TI

)≥ 0 ∀ ∆ ∈ ∆

if ( and only if )

1) ∃ Φ ∈ Φ∆

2)(

MI

)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0

0 Y∗ 0 Z

(MI

)> 0







)∗ ( X YY∗ Z

)(TI

)≥ 0 ∀ ∆ ∈ ∆

if ( and only if )

1) ∃ Φ ∈ Φ∆

2)(

MI

)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0

0 Y∗ 0 Z

(MI

)> 0

Local step : find simple QC for every Ti







)∗ ( X YY∗ Z

)(TI

)≥ 0 ∀ ∆ ∈ ∆

if ( and only if )

1) ∃ Φ ∈ Φ∆

2)(

MI

)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0

0 Y∗ 0 Z

(MI

)> 0

Local step : find simple QC for every Ti =⇒ reduce the complexity







)∗ ( X YY∗ Z

)(TI

)≥ 0 ∀ ∆ ∈ ∆

if ( and only if )

1) ∃ Φ ∈ Φ∆

2)(

MI

)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0

0 Y∗ 0 Z

(MI

)> 0


Ti are seen as uncertainty ∆i







)∗ ( X YY∗ Z

)(TI

)≥ 0 ∀ ∆ ∈ ∆

if ( and only if )

1) ∃ Φ ∈ Φ∆

2)(

MI

)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0

0 Y∗ 0 Z

(MI

)> 0



Global step : use local QC to find global QC







)∗ ( X YY∗ Z

)(TI

)≥ 0 ∀ ∆ ∈ ∆

if ( and only if )

1) ∃ Φ ∈ Φ∆

2)(

MI

)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0

0 Y∗ 0 Z

(MI

)> 0



Global step : use local QC to find global QC =⇒ conservative results







)∗ ( X YY∗ Z

)(TI

)≥ 0 ∀ ∆ ∈ ∆

if ( and only if )

1) ∃ Φ ∈ Φ∆

2)(

MI

)∗ −Φ22 0 −Φ∗12 00 X 0 Y−Φ12 0 −Φ11 0

0 Y∗ 0 Z

(MI

)> 0



Global step : use local QC to find global QC =⇒ conservative results

=⇒ create a basis for QC of Ti (to use as Φ∆ in global step)




Classical interpretation :For given X, Y and Z find Φ from basis Φ∆

New interpretation :

Find basis for X, Y and Z from given Φ ∈ Φ∆

Propagate the old basis into the new basis

=⇒ QC propagation




Classical interpretation :For given X, Y and Z find Φ from basis Φ∆

New interpretation :

Find basis for X, Y and Z from given Φ ∈ Φ∆

Propagate the old basis into the new basis

=⇒ QC propagation

Difficulties

Size : not too big/small

Quality : describes the best the uncertain system

Efficient computation : convex



Robustness Analysis : QC classes

Some classes of QC with geometric interpretationsdisc [Dinh et al., 2013]band [Dinh et al., 2014]cone [Laib et al., 2015]

Real(w*z)0.98 0.99 1 1.01 1.02 1.03

Imag

inar

y (w

*z)

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

Real(w*z)0.98 0.99 1 1.01 1.02 1.03

Imag

inar

y (w

*z)

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

Real(w*z)0.98 0.99 1 1.01 1.02 1.03

Imag

inar

y (w

*z)

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

Formulate as convex optimization (no graphical computation)

Some physical interests : gain, phase, . . .





Real(w*z)0.98 0.99 1 1.01 1.02 1.03

Imag

inar

y (w

*z)

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

Real(w*z)0.98 0.99 1 1.01 1.02 1.03

Imag

inar

y (w

*z)

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

Real(w*z)0.98 0.99 1 1.01 1.02 1.03

Imag

inar

y (w

*z)

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0



Cone : Phase uncertainty information

The phase notion for Single-Input Single-Output (SISO) systems is well defined

For Multi-Input Multi-Output (MIMO) systems ? ?





Real(w*z)0.98 0.99 1 1.01 1.02 1.03

Imag

inar

y (w

*z)

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

Real(w*z)0.98 0.99 1 1.01 1.02 1.03

Imag

inar

y (w

*z)

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

Real(w*z)0.98 0.99 1 1.01 1.02 1.03

Imag

inar

y (w

*z)

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0



Cone : Phase uncertainty information

The phase notion for Single-Input Single-Output (SISO) systems is well defined

For Multi-Input Multi-Output (MIMO) systems ? ?=⇒ Numerical range



Robustness Analysis : Numerical Range

For a given a frequency response Γ of a system T

The numerical range N (Γ)

N (Γ) = w∗z | z = Γw,w ∈ Cnw and ‖w‖ = 1







Certain numerical range







Certain numerical range Uncertain numerical range



Robustness Analysis : Cone QC [Laib et al., 2015]

Theorem

Given the frequency responseof an uncertain system T

Finding the smallest α :




Theorem

Given the frequency responseof an uncertain system T

Finding the smallest α :

Quasiconvex optimisation problem

LMI constraints




Theorem

Given T = ∆ ?M, let λ = cotα

2

minλ,Ω

D1, G1, D1, G1

D2, G2, D2, G2

λ

s.t :

λ

(D1 0

0 D2

)+

(D1 0

0 −D2

)> 0

λ

(M 0I 00 M0 I

)∗ (B1 00 B2

)(M 0I 00 M0 I

)+

(M 0I 00 M0 I

)∗ (A1 00 A2

)(M 0I 00 M0 I

)> 0(

D1 0

0 D2

)> 0 and

(M 0I 00 M0 I

)∗ (B1 00 B2

)(M 0I 00 M0 I

)> 0

=⇒ Efficient tools to solve the problem


Proposed approach Hierarchical approach

Resume

Level 1

pq

M

T

T1

z w

Ti

TN

1

1 1

11

1

1

1

1

1 Consider hierarchical structure of the systemFind basis (QC description) for Ti with Robust Performance TheoremPropagate this basis to the global level

2 For global hierarchical level, investigate the performance with RobustPerformance Theorem



Resume

Level 1

pq

M

T

T1

z w

Ti

TN

1

1 1

11

1

1

1

1

1 Consider hierarchical structure of the system

Find basis (QC description) for Ti with Robust Performance TheoremPropagate this basis to the global level




Resume

Level 1

pq

M

T

T1

z w

Ti

TN

1

1 1

11

1

1

1

1

sTi

uiyi

qipi

Level 2

∆i

M11

M21 M22

M12i i

ii

1 Consider hierarchical structure of the system

Find basis (QC description) for Ti with Robust Performance TheoremPropagate this basis to the global level




Resume

Level 1

pq

M

T

T1

z w

Ti

TN

1

1 1

11

1

1

1

1

sTi

uiyi

qipi

Level 2

∆i

M11

M21 M22

M12i i

ii

1 Consider hierarchical structure of the systemFind basis (QC description) for Ti with Robust Performance Theorem

Propagate this basis to the global level




Resume

Level 1

pq

M

T

T1

z w

Ti

TN

1

1 1

11

1

1

1

1

sTi

uiyi

qipi

Level 2

∆i

M11

M21 M22

M12i i

ii





Resume

Level 1

pq

M

∆

∆1

z w

∆i

∆N

1

1 1

11

u1

u1

u1

u1

sTi

uiyi

qipi

Level 2

∆i

M11

M21 M22

M12i i

ii





Resume

Level 1

pq

M

∆

∆1

z w

∆i

∆N

1

1 1

11

u1

u1

u1

u1

sTi

uiyi

qipi

Level 2

∆i

M11

M21 M22

M12i i

ii

Computation time is reduced however conservatism may appear

robustness of feedbacks loops =⇒ simple set may be sufficient

combination of several simple sets =⇒ increase of the computation time



Resume

Level 1

pq

M

∆

∆1

z w

∆i

∆N

1

1 1

11

u1

u1

u1

u1

sTi

uiyi

qipi

Level 2

∆i

M11

M21 M22

M12i i

ii

Computation time is reduced however conservatism may appear

robustness of feedbacks loops =⇒ simple set may be sufficient

combination of several simple sets =⇒ increase of the computation time

=⇒ trade-off conservatism/computation time


Application Example

PLL network : Local Step

Characterize each PLL with QC with : disc, band and cone

Real(w*z)0.9998 0.9999 1 1.0001 1.0002 1.0003 1.0004

Imag

inar

y (w

*z)

×10-4

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

ω = 15

Real(w*z)0.999 0.9995 1 1.0005 1.001 1.0015 1.002

Imag

inar

y (w

*z)

×10-3

-1.5

-1

-0.5

0

0.5

1

ω = 37

Real(w*z)0.99 0.995 1 1.005 1.01 1.015

Imag

inar

y (w

*z)

×10-3

-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

ω = 136

Real(w*z)0.95 1 1.05

Imag

inar

y (w

*z)

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

ω = 501


Application Example

PLL network : Global Step

Compute an upper bound on Trg→eg for all the uncertainties

ω, rad/sec101 102 103 104

Mag

nit

ud

e, d

B

-40

-30

-20

-10

0

10

20PLL network performance analysis

Direct approach: 346.7 sec


Application Example



ω, rad/sec101 102 103 104

Mag

nit

ud

e, d

B

-40

-30

-20

-10

0

10



Hierarchical approach disc: 16 sec


Application Example



ω, rad/sec101 102 103 104

Mag

nit

ud

e, d

B

-40

-30

-20

-10

0

10




Hierarchical approach disc+band: 46.4 sec


Application Example



ω, rad/sec101 102 103 104

Mag

nit

ud

e, d

B

-40

-30

-20

-10

0

10




Hierarchical approach disc+band: 46.4 sec

Hierarchical approach disc+cone: 43.1 sec


Discussion

General Hierarchical Approach

Hierarchical approach

Level 1

pq

M

∆

∆1

z w

∆i

∆N

1

1 1

11

u1

u1

u1

u1

sTi

uiyi

qipi

Level 2

∆i

M11

M21 M22

M12i i

ii


Discussion


Hierarchical approach

Level 1

pq

M

∆

∆1

z w

∆i

∆N

1

1 1

11

u1

u1

u1

u1

sTi

uiyi

qipi

Level 2

∆i

M11

M21 M22

M12i i

ii

Special case : Direct approach

Level 1

pq

M

∆

∆1

z w

∆i

∆N

1

1 1

11

u1

u1

u1

u1

sTi

uiyi

qipi

Level 2

∆i

M11

M21 M22

M12i i

ii


Discussion


pq

M

T

T1

z w

Ti

TN

Level 1

1

1 1

11

1

1

1

1


Discussion


pq

M

T

T1

z w

Ti

TN

Level 1

1

1 1

11

1

1

1

1

Level 2

pq

M

T

z w

Ti

TN

2

2 2

22

2

2

2

2

T1


Discussion


pq

M

T

T1

z w

Ti

TN

Level 1

1

1 1

11

1

1

1

1

Level 2

pq

M

T

z w

Ti

TN

2

2 2

22

2

2

2

2

T1s

Ti

uiyi

qipi

l

Level l

∆i

M11

M21 M22

M12i i

ii


Discussion


pq

M

T

T1

z w

Ti

TN

Level 1

1

1 1

11

1

1

1

1

Level 2

pq

Mz w2

2 2

22

∆

∆1

∆i

∆Nu2

u2

u2

u2

sTi

uiyi

qipi

l

Level l

∆i

M11

M21 M22

M12i i

ii


Discussion


pq

M

Δ

Δ 1

z w

Δ

Δ N

Level 1

1

1 1

11

u1

u1

u1

u1

Level 2

pq

Mz w2

2 2

22

∆

∆1

∆i

∆Nu2

u2

u2

u2

sTi

uiyi

qipi

l

Level l

∆i

M11

M21 M22

M12i i

ii


Discussion


pq

M

Δ

Δ 1

z w

Δ

Δ N

Level 1

1

1 1

11

u1

u1

u1

u1

Level 2

pq

Mz w2

2 2

22

∆

∆1

∆i

∆Nu2

u2

u2

u2

sTi

uiyi

qipi

l

Level l

∆i

M11

M21 M22

M12i i

ii

Many degrees of freedom to handle the trade-off conservatism/computation time


Discussion


pq

M

Δ

Δ 1

z w

Δ

Δ N

Level 1

1

1 1

11

u1

u1

u1

u1

Level 2

pq

Mz w2

2 2

22

∆

∆1

∆i

∆Nu2

u2

u2

u2

sTi

uiyi

qipi

l

Level l

∆i

M11

M21 M22

M12i i

ii

Many degrees of freedom to handle the trade-off conservatism/computation time

Number of levels

Number of Ti in each level

Basis for ∆i

Basis for Ti in each level

Parallel computing


Discussion

Robust stability

Network with N systems randomly generated [Andersen et al., 2014].


Discussion

Robust stability


Complex large scale

analysis problem

Complex large scale


Small scale



High complexity

Low complexity

Specific solver



Direct method computation timeProposed method computation time

= 10 for N = 200


Discussion

Robust stability


Complex large scale

analysis problem

Complex large scale


Small scale



analysis problems

(distributed)


High complexity

Low complexity

Specific solver





solver

N0 50 100 150 200

CP

U t

ime

(sec

on

ds)

10-2

10-1

100

101

102

Direct methodDistributed hierarchical method

Direct method computation timeProposed method computation time

= 113 for N = 200


Conclusion and future work

Conclusion

Performance analysis of uncertain large scale systems

Important computation time with direct method

Exploit hierarchical structure using basis (QC) propagation

General approach with degrees of freedom

Reduce computation time with possible conservatism

Trade-off conservatism/computation time



Perspectives

Perspectives

Systematic decomposition technique using Graph Theory

Combine hierarchical method with specific solvers

Complex large scale

analysis problem

Complex large scale


Small scale



analysis problems

(distributed)


High complexity

Low complexity

Specific solver





solver



Thank you for your attention



References

Andersen, M., Pakazad, S., Hanson, A., and Rantzer, A. (2014).

Robust stability analysis of sparsely interconnected uncertain systems.IEEE Transactions on Automatic Control, 59(8) :2151–2156.

Dinh, M., Korniienko, A., and Scorletti, G. (2013).

Embedding of uncertainty propagation : application to hierarchical performance analysis.IFAC Symposium on System, Structure and Control, 5(1) :190–195.

Dinh, M., Korniienko, A., and Scorletti, G. (2014).

Convex hierarchicalrchical analysis for the performance of uncertain large scale systems.IEEE Conference on Decision and Control, pages 5979– 5984.

Laib, K., Korniienko, A., Scorletti, G., and Morel, F. (2015).

Phase IQC for the hierarchical performance analysis of uncertain large scale systems.IEEE Conference on Decision and Control (to appear).


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